FINITE GROUP THEORY. FOR COMBINATORISTS Volume one. Jin Ho Kwak. Department of Mathematics POSTECH Pohang, Korea

Size: px
Start display at page:

Download "FINITE GROUP THEORY. FOR COMBINATORISTS Volume one. Jin Ho Kwak. Department of Mathematics POSTECH Pohang, 790 784 Korea jinkwak@postech.ac."

Transcription

1 FINITE GROUP THEORY FOR COMBINATORISTS Volume one Jin Ho Kwak Department of Mathematics POSTECH Pohang, Korea Ming Yao Xu Department of Mathematics Peking University Beijing , P.R. China This work is supported by Com 2 MaC-KOSEF, Korea.

2 ii

3 Preface This book is designed for a text on finite group theory for combinatorists, mainly for graph-theorists. It has three parts. This volume is the first part which gives the basic concepts and theorems on abstract groups. The volume 2 contains the second part which is a text and also a reference book on permutation groups. The volume 3 contains the third part which gives some examples for applying group theory to the combinatorial problems. For volume 1, the second author used it three times as a text on group theory for graduate students. The first one was in 2000 at a summer school for graduate students held in Kunming, China. The second time was in spring semester in 2001 at Temple University, the United States. The third time was in spring semester in 2005 at Pohang University of Science and Technology, Korea. The present version is a expansion of his lecture notes. We would take this opportunity to thank all people who give us valuable comments and suggestions for revising this book, including many students. Especially we are grateful to Professor Seymour Lipschutz, at Temple University, who read the manuscript carefully and corrected some errors and polished the English. We also thank Shanxi Teachers University, China, for its hospitality; part of work of writing and revising this book was done when the second author visited there. Finally we are grateful to the Combinatorial and Computational Mathematics Center at Pohang University of Science and Technology, Korea, for accepting this book as one of the Lecture Note Series. iii Jin Ho Kwak Ming-Yao Xu July 2005, in Pohang, Korea

4

5 Contents Preface iii 1 Basic concepts Groups and subgroups Normal subgroups and direct products Examples of groups Commutators and solvable groups Automorphisms Some miscellaneous properties Free groups, generators and relations Graphs of groups and Groups of graphs Group actions Group actions on a set Sylow Theorems p-groups and solvable groups Transitive permutation representations Transfer and Burnside Theorem The structure theory The Jordan-Hölder Theorem Decomposition into direct products Extensions and Schur-Zassenhaus Theorem Classifying groups of special orders Wreath products v

6 vi 4 Nilpotent groups and solvable groups Commutators Nilpotent groups Frattini subgroups p-groups Hall s enumeration principle Solvable groups Fitting subgroups Sylow basis for solvable groups Representations of finite groups Representations of groups Characters of groups Induced representations Applications Selected Answers and Hints 221 Bibliography 247 Index 255

7 vii

8 viii Notation Notation N, Z natural numbers and integers Q, R, C rational, real and complex numbers (m, n) greatest common divisor of m and n G, H,... sets, groups, etc X, D,... classes of groups α, β, γ,... functions x, y, z,... elements of a set x α image of x under α G α image of G under α Ker α kernel of α x y y 1 xy Cl(g) the conjugacy class containing g [x, y] x 1 y 1 xy H = G H is isomorphic with G H char G H is a characteristic subgroup of G H G H is isomorphic to a subgroup of G H G, H < G H is a subgroup, a proper subgroup of the group G H G, H G H is a normal subgroup, a proper normal subgroup of G H sn G H is a subnormal subgroup of G H 1 H 2 H n product of subsets of a group X λ λ Λ subgroup generated by subsets X λ of a group X R group presented by generators X and relators R M subgroup generated by M N 1, N 2,..., N s subgroup generated by N 1, N 2,..., N s d(g) minimum number of generators of G G n subgroup generated by all g n where g G S cardinality of the set S G : H index of the subgroup H in the group G o(x) order of the group element x

9 Notation ix C G (H), N G (H) H G H g H G or Core G (H) Aut(G), Inn(G) Out(G) Hol(G) Hom Ω (G, H) End Ω (G) H 1 H n, H 1 H n N H H K H K G = [G, G] G (i) G i, Z i (G) k(g) Z(G) F (G) Φ(G) M(G) O π (G) l π (G) S X S n, A n D 2n Q 2 n Z n R RG GL(V ) GL(n, F), SL(n, F) PGL(n, F), PSL(n, F) centralizer, normalizer of H in G normal closure of H in G conjugate of H by g core of M in G automorphism group, inner automorphism group of G Aut(G)/Inn(G), outer automorphism group of G holomorph of G set of Ω-homomorphisms from G to H set of Ω-endomorphisms of G set product direct products, direct sums semidirect products wreath products tensor product derived subgroup of a group G term of the derived series of G terms of the lower central series, the upper central series of G the number of conjugacy classes of G center of G Fitting subgroup of G Frattini subgroup of G Schur multiplicator of G maximal normal π-subgroup of G π-length of G symmetric group on X symmetric, alternating groups of degree n dihedral group of order n generalized quaternion group of order 2 n Z/nZ, cyclic group of order n group of units of a ring R with identity group ring of a group G over a ring R with identity element group of nonsingular linear transformations of a vector space V general linear and special linear linear groups projective general linear and projective special linear groups

10 x Notation M G, χ G max, min E ij F q, GF (q) F d, V (n, F) AG(d, F), AG(d, q) PG(d, F), PG(d, q) S(t, k, v) S n, A n GL(d, F), SL(d, F), ΓL(d, F) GL(d, q), SL(d, q), ΓL(d, q) AGL(d, F), ASL(d, F), AΓL(d, F) AGL(d, q), ASL(d, q), AΓL(d, q) PGL(d, F), PSL(d, F), PΓL(d, F) PGL(d, q), PSL(d, q), PΓL(d, q) Sp(2m, F) Sp(2m, q) PGU(d, F), PSU(d, F), PΓU(d, F) PGU(d, q), PSU(d, q), PΓU(d, q) Sz(2 s ) and R(3 s ) M 10,..., M 24 W 10,..., W 24 fix(x), supp(x) Ω {k}, Ω (k) x N G (H) G H, G m G.H G : H induced module, induced character maximal, minimal conditions matrix with (i, j) entry 1 and other entries 0 field with q elements vector space of dimension n over F affine geometry over F and over F q projective geometry over F and over F q Steiner system symmetric and alternating groups of degree n linear groups over F linear groups over GF (q) affine groups over F affine groups over GF (q) projective groups over F projective groups over GF (q) sympletic groups over F sympletic groups over GF (q) unitary groups over F unitary groups over GF (q) Suzuki and Ree groups Mathieu groups Witt geometries set of fixed points and support of x sets of k-subsets and k-tuples from Ω largest integer x socle of G direct product, direct power an extension of G by H a split extension of G by H

11 Volume one Abstract Groups

12

13 Chapter 1 Basic concepts We begin with some basic definitions and elementary properties of finite groups, up to but not including Sylow s Theorems. Most of the material in this chapter can be found in lots of textbooks of abstract algebra; so we omit a proof in many cases. 1.1 Groups and subgroups Definition A nonempty set G with a closed binary operation called a multiplication is called a monoid if the following two axioms hold: (1) Associative law: (ab)c = a(bc) for all a, b, c G; (2) Identity element: there exists 1 G such that 1a = a1 = a for all a G. A monoid is called a group if (3) Inverse element: for each a G, there is an element a 1 G such that aa 1 = a 1 a = 1. A group G is called abelian (or commutative) if (4) Commutative law: ab = ba for all a, b G. We use 1 to denote both the identity element in a group and the group consisting of the identity element only. We use G to denote the 3

14 4 Basic concepts cardinality of a set G, and if G is a group we call G the order of G. The group G is called finite if G is a positive integer; otherwise G is called infinite. Elementary examples of groups come from number systems. The complex number system C is a group under ordinary addition; but not a group under ordinary multiplication because 0 has no inverse. However, the nonzero complex numbers C \ {0} form a group under ordinary multiplication. As subsets, the integers Z, the rational numbers Q, and the real numbers R are groups under addition, and the nonzeros of Q and R form groups under multiplication but Z \ {0} is not. Note that all mentioned groups are abelian. For another example, let M = {1, 2,..., n} and let S M be the set of all bijections M M, called permutations. Under the usual composition of functions, 1 S M becomes a group, called the symmetric group on n elements and denoted S n. Note that S n = n! and S n is not abelian for n 3. Throughout this book, all groups are assumed to be finite unless otherwise stated. By the associative law, products of any finite number of elements of G in a certain order are meaningful. Thus one may define the powers of an element a of G as follows: For any positive integer n, Obviously we have a n = aa } {{ a}, a 0 = 1, a n = (a 1 ) n. n a m a n = a m+n and (a m ) n = a mn for any integers m, n. Let H and K be subsets of a group G. We define a product HK := {hk h H, k K}. (1.1) 1 For any σ, τ S M, one can define their multiplication σ τ as either σ τ or τ σ, where (σ τ)(x) = σ(τ(x)). In the former case, we write (σ τ)(x) = (σ τ)(x) like the usual function notation. However, in the latter case, we use x σ instead of the usual notation σ(x) to get x σ τ = (x σ ) τ. Throughout this book, the latter case is preferred.

15 Groups and subgroups 5 Since G is associative, so is this product operation. If K = {a} then we write Ha for H{a}; similarly we write ah for {a}h. Also we define H 1 = {h 1 h H}; and for any positive integer n, define H n = {h 1 h 2 h n h i H}. Definition A nonempty subset H of a group G is called a subgroup of G, denoted by H G, if H 2 H and H 1 H. It is easy to see that if H is a subgroup of G then H 2 = H, H 1 = H and 1 H, and hence H satisfies the axioms of a group. Obviously, every group G 1 has at least two subgroups, 1 and G itself; 1 is called the trivial subgroup of G. Sometimes, we write H < G if H G but H G. Proposition Let G be a group and H G a nonempty subset. Then the following statements are equivalent: (1) H G; (2) ab H and a 1 H for any a, b H; (3) ab 1 H (or a 1 b H) for any a, b H. (4) H 2 H (if G is finite). The intersection of a collection of subgroups of a group G is also a subgroup of G, but the union of several subgroups is not necessarily a subgroup. Definition Let G be a group and M G. The intersection of all subgroups of G containing M is called the subgroup generated by M, denoted by M. It is easy to see that M = {1, a 1 a 2 a n a i M M 1, n = 1, 2,...}.

16 6 Basic concepts If M = G, M is said to be a generating set of G or G is generated by M. A group generated by one element is called a cyclic group. A group generated by a finite number of elements is called a finitely generated group. For an element a in a group G, we call the order of the subgroup a the order of a, denoted by o(a), that is o(a) = a. One can show that o(a) is the smallest positive integer n satisfying a n = 1; and if such an integer n does not exist we say o(a) =. For a group G, we call the least common multiple of the orders of elements of G the exponent of G, denoted by exp G. Note that for any two subgroups H and K of a group G, their product HK is not necessary to be a subgroup. However, we have the following Theorem Let G be a group and H G, K G. Then HK G HK = KH. Proof: ( ) First, note that for any subgroup L, we have L = L 1. Assume that HK G. Then HK = (HK) 1 = K 1 H 1 = KH. ( ) Assume that HK = KH. Then (HK) 2 = HKHK = HHKK = HK, (HK) 1 = K 1 H 1 = KH = HK. By Definition 1.1.2, we have HK G. Example Let G be the symmetry group of the regular tetrahedron having vertices 1, 2, 3, 4. The order of G is 12. In fact, it consists of the even permutations of four vertices 1, 2, 3, 4, and then G = A 4, the alternating group acting on the set {1, 2, 3, 4}. Let H = (123) and K = (12)(34). Since HK KH, HK is not a subgroup of G. In fact, HK = 6, but A 4 has no subgroup of order 6. (A 4 is also called the tetrahedron group. (Hint: if it had, then it would be normal; the intersection of this subgroup and the normal subgroup of order 4 would be a normal subgroup of order 2, impossible.) (The alternating group will be defined later in page 28.)

17 Groups and subgroups 7 Definition Let H G and a G. The subset ah (or Ha, resp.) is called a left coset (or right coset, resp.) of H. It is easy to see that ah = bh a 1 b H, while Ha = Hb ab 1 H. Proposition Let H G and let a, b G. Then (1) ah = bh ; (2) If ah bh then ah = bh. Now, the group G can be expressed as a disjoint union of the left cosets of H: G = a 1 H a 2 H a n H, where {a 1, a 2,..., a n }, the set of representatives of the left cosets of H in G, is called a transversal of the coset decomposition. The number n of distinct left cosets of H in G is called the index of H in G, denoted by G : H, which is not necessarily finite, The same conclusion is true for right cosets. So, G : H is also the number of right cosets of H in G. The following theorem is a basic property of subgroups of a finite group. Theorem (Lagrange) Let G be a group and let H G. Then G = H G : H. For a group G, the order of a subgroup of G is a divisor of G by Lagrange s Theorem In particular, the order o(a) of any element a is a divisor of G. The converse of Lagrange s Theorem does not hold, as shown in Example One might ask under what condition, the converse can hold. A partial answer will be mentioned later as Sylow Theorem, and the converse holds for any abelian or nilpotent group. (See Theorem1.2.9 and Corollary ) Also, the converse of Lagrange s Theorem holds for the symmetric group S 4 ; the verification will be left to the reader. For further information see Exercise

18 8 Basic concepts Theorem Let G be a group, and let H and K be two subgroups of G. Then HK = H K H K. Proof: Since HK is a disjoint union of right cosets of the form Hk, k K, and every right coset has H elements, it suffices to show that HK contains K : H K right cosets of H in G. It is easy to see that Noting that k 1 k 1 2 K, we have Hk 1 = Hk 2 k 1 k 1 2 H. Hk 1 = Hk 2 k 1 k2 1 H K (H K)k 1 = (H K)k 2. It follows that the number of right cosets of H in HK is equal to the index of H K in K, i.e., to K : H K. Proposition Let G be a group and let H G and K G. Then (1) H, K : H K : H K ; (2) G : H K = G : H H : H K G : H G : K ; (3) If G : H and G : K are relatively prime, then G : H K = G : H G : K and G = HK. Proof: (1) From the proof of Theorem 1.1.5, there are KH : H = K : H K right cosets of H in HK, here we slightly abuse to use the notation KH : H because KH is not necessarily a subgroup of G; in fact, we use KH : H to denote the number of cosets of H in KH. Since H, K HK, we have H, K : H HK : H = K : H K. (2) Since G : H K = G : K K : H K,

19 Groups and subgroups 9 and G : H H, K : H, we have G : H K : H K by (1), and hence G : H K G : H G : K. (3) By Lagrange s Theorem, G : H and G : K are divisors of G : H K. Since G : H and G : K are relatively prime, we have It follows from (2) that On the other hand, G : H G : K is a divisor of G : H K. G : H K = G : H G : K. G : H K = G : K K : H K = G : K HK : H, and hence G : H = HK : H, and G = HK. Let G be a group, and let a, g G. We write a g = g 1 ag. We call a g the conjugate of a by g. For a subgroup or a subset H of G, let H g = g 1 Hg, which is also called the conjugate of H by g. Two elements a and b (or two subgroups or subsets H and K) of G are called conjugate in G if there is an element g G such that a g = b (or H g = K). It is easy to see that the conjugacy relation between elements (or subgroups or subsets) is an equivalence relation. The corresponding equivalence classes C 1 = {1}, C 2,..., C k, are called the conjugacy classes of G, also we have G = C 1 C 2 C k (a disjoint union), and hence G = C 1 + C C k. The latter equation is called the class equation of G. The number of elements in C i, denoted by C i, is called the length of C i.

20 10 Basic concepts Definition Let G be a group and let H be a subset (or a subgroup) of G. (1) An element g G normalizes H if H g = H. We call the normalizer of H in G. N G (H) = {g G H g = H} (2) An element g G centralizes H if h g = h for any h H. We call C G (H) = {g G h g = h, h H} the centralizer of H in G. If H = G, then Z(G) := C G (G) is called the center of G. Note that for a subgroup H of G, C G (H) = G if and only if H Z(G). It is easy to see that, for any subset H, N G (H) and C G (H) are subgroups of G with N G (H) C G (H), and that if H G then H N G (H). When H = {a} is a one-element set, we write N G (a) and C G (a) for N G (H) and C G (H), respectively. In this case, we have C G (a) = N G (a). Theorem Let G be a finite group. Then, (1) the conjugacy class C of G containing an element a has length C = G : C G (a), and hence C is a divisor of G. (2) The number of subgroups (or subsets) conjugate to a subgroup (or a subset) H equals the index G : N G (H) of the normalizer N G (H) in G, which is a divisor of G. Note that a conjugate of a subgroup H of G is a different concept from a conjugacy class of G. In fact, the conjugacy classes of G form a partition of a group G. However, the conjugates {H g g G} of H are neither disjoint each other nor a cover of G. (See Proposition ) Next, we generalize the concept of right or left cosets. Definition Let H and K be two (not necessarily distinct) subgroups of a group G. The subset HaK, a G, is called a double coset of G with respect to H and K, or simply an (H, K)-double coset.

21 Groups and subgroups 11 Similar to cosets, we have Proposition For any a, b G, HaK HbK HaK = HbK. So G can be decomposed into a disjoint union of double cosets: G = Ha 1 K Ha 2 K Ha s K. The double coset HaK is a union of several right cosets of H, and also is a union of several left cosets of K. Theorem Let G be a group and let H and K be any two (not necessarily distinct) subgroups of G. Then the number of right cosets of H in double coset HaK is K / H a K, and the number of left cosets of K in HaK is H / H K a. Proof: We only prove the first statement. The number of right cosets of H in HaK is HaK / H. Since HaK = a 1 HaK = H a K, by Theorem 1.1.5, we have HaK = H a K / H a K = H K / H a K, from which the conclusion follows. Let G and H be two groups. A map α : G H is called a homomorphism of G into H if (ab) α = a α b α, a, b G. If a homomorphism α is surjective (or injective), then we call α an epimorphism (or monomorphism); if α is a bijection, then α is an isomorphism from G onto H. In this case, G and H are said to be isomorphic, denoted by G = H. A homomorphism and an isomorphism from G to itself is called an endomorphism and an automorphism of G, respectively. We use End(G) to denote the set of all endomorphisms of G, and Aut(G) the set of all automorphisms of G. For the composition of maps, End(G) is a monoid, and Aut(G) is a group, called the automorphism group of G.

22 12 Basic concepts For g G, the map σ(g) : G G defined by a σ(g) = a g, a G, is an automorphism of G, which is called the inner automorphism of G induced by g, denoted by Inn(g). The set Inn(G) of all inner automorphisms of G is a subgroup of Aut(G), called the inner automorphism group of G. The map σ : g Inn(g) is an epimorphism from G to Inn(G). Exercises Let G be a group, and let g G, o(g) = n. Then o(g m ) = n/(m, n) Let H G, g G. If o(g) = n and g m H, (n, m) = 1, then g H Let G be a group. If exp G = 2, then G is an abelian group Let H G, K G, and a, b G. If Ha = Kb, then H = K Let A, C be subgroups of G. If A C, then AB C = A(B C) for any subgroup B of G Let A, B, C be subgroups of G, and let A B. If A C = B C and AC = BC, then A = B Prove that a group having no nontrivial subgroups is cyclic of prime order Determine the conjugacy classes of A 4 and S (The number of conjugacy classes in G) The number of conjugacy classes in a finite group G is called the class number of G, denoted by k(g). E. Landau(1903) answered an open question posed by Frobenius by showing that for any positive integer k, there are only a finite number of finite groups G with k(g) = k. In relation to the class number k(g), let n(k) denote the maximum order of a group having k classes, and k(n) the minimum number of classes in a group of order n. Show that: (i) Prove that n(1) = 1, n(2) = 2, n(3) = 6, n(4) = 12 and n(5) = 60. (ii) For small k determine all finite groups with class number k. (iii) For a given n, find a lower bound for k(n); or equivalently, for a given k, find an upper bound for n(k). Landau proved n(k) k 2k, see Brauer (1963). For estimating k(n), see also Ayoub (1967), Sherman (1969) and Poland (1968) Show Aut(S n ) = S n except n = 6. Also, determine Aut(S 6 ). Furthermore, show that Aut(A n ) = Aut(S n ).

23 Normal subgroups and direct products Show that A n has a unique conjugacy class of subgroups isomorphic to A n 1 if n 6; and has exactly two such classes if n = Given a natural number n, it is one of the basic problems in group theory to determine all non-isomorphic groups of order n. With the aid of a computer, Besche, Eick and O Brien (2002) determined all groups of order The number of non-isomorphic groups of order n are 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, etc. For what n you can solve the same problem by hand now, or after studying the first three chapters of this book? For more information, visit The Small Groups library at hubesche/small.html. 1.2 Normal subgroups and direct products We study a subgroup of a group whose left cosets coincide with the right cosets. Definition A subgroup N of a group G is called normal, denoted by N G if N g N, g G. Obviously, every subgroup of an abelian group is normal. And any group G has at least two normal subgroups, that is G and 1, called the trivial normal subgroups. Proposition Let G be a group. Then the following statements are equivalent: (1) N G; (2) N g = N, g G (say, a self-conjugate subgroup); (3) N G (N) = G; (4) For each g N, we have Cl(g) N, where Cl(g) denotes the conjugacy class of G containing g. (5) Ng = gn, g G, that is, every left coset of N in G is also a right coset.

24 14 Basic concepts Definition A group is called simple if it has no nontrivial normal subgroups. Since each subgroup of an abelian group is normal, the only abelian simple groups are cyclic of prime order, (see Exercise 1.1.7). However, a classification of all nonabelian simple groups was one of long standing open problems in finite group theory. Although in Galois age some nonabelian simple groups, say A n for n 5, were known, all finite simple groups have been found in early 1980s. There are 18 infinite families of finite simple groups and 26 sporadic simple groups in totality. The classification theorem of finite simple groups asserts that every finite simple group is isomorphic to one of groups in the 18 families or one of the 26 sporadic groups. The proof of this theorem was scattered in about 500 papers. A readable proof is still in preparation. Very recently, M. Aschbacher and S. Smith successfully classified so-called finite quasithin simple groups in 2003, the classification of finite simple groups is finally completed. For more information we refer the reader to the well-organized book, Gorenstein (1982). Also, we shall give more detailed description for finite simple groups in volume two. It was shown that for two subgroups H and K of a group G, HK is not necessary to be a subgroup, but it is a subgroup if at least one of H and K is normal, by Theorem The next proposition shows that it will be a normal subgroup if both H and K are normal. Proposition Let N 1, N 2,..., N s be normal subgroups of G. Then s i=1 N i and N 1, N 2,..., N s are also normal, and N 1, N 2,..., N s = N 1 N 2 N s. Definition Let G be a group and M G. We call M G = m g m M, g G the normal closure of M in G or the normal subgroup generated by M. It is easy to see that the normal closure M G consists of elements of the form l k=1 g 1 k m ε k k g k, g k G, m k M, ε k = ±1,

25 Normal subgroups and direct products 15 and hence M G is the smallest normal subgroup of G containing M. Assume that N G. Consider the set G = {Ng g G} of all (right) cosets of N in G. Define a multiplication in G as shown in Eq.(1.1), that is, the multiplication of subsets in G, namely, (Ng)(Nh) = N(gN)h = N(Ng)h = N 2 gh = Ngh. (1.2) Then G = {Ng g G} is a group with this multiplication. (The converse is also true. That is, if G is a group with the multiplication defined in Eq.(1.2), then N G.) It is called the factor group or quotient group of G modulo N, and denoted by G = G/N. For any positive integer n, nz is a normal subgroup of the integer group Z. The factor group Z n = Z/nZ = {0, 1,..., n 1} is called the (additive) group of integers modulo n. In this group, (nz)a (nz)b = (nz)c if and only a + b = c (mod n). Let α : G H be a group homomorphism. Then is called the kernel of α, and Ker α = {g G g α = 1} G α = {g α g G} is called the image of α. It is easy to see that Ker α G and G α H. Theorem (The Fundamental Theorem of Homomorphisms) (1) Let N G. Then the map ν : g Ng is a homomorphism of G onto G/N with Ker ν = N and G ν = G/N. Such a homomorphism ν is called the natural homomorphism from G onto G/N. (2) Let α : G H be a homomorphism. Then Ker α G and G α = G/Ker α. The next two theorems are the first and the second theorems of isomorphisms. Their proofs can be found in most textbooks of abstract algebra.

26 16 Basic concepts Theorem (The First Isomorphism Theorem) Let N G, M G and N M. Then M/N G/N and (G/N)/(M/N) = G/M. Theorem (The Second Isomorphism Theorem) G and N G. Then Let H (H N) H and HN/N = H/(H N). Given two groups G, H, their (outer) direct product is defined by G H = {(g, h) g G, h G}, where the multiplication is defined by (g, h)(g, h ) = (gg, hh ) for g, g G, h, h H. In the same way, one may define the (outer) direct product of n groups G 1,..., G n. Let G be the (outer) direct product of n groups G 1,..., G n : For i = 1, 2,..., n, let G = G 1 G n. H i = {(1,..., 1, g i, 1,..., 1) g i G i }, where g i is the i-th component. Then H i = Gi, and the following hold: (1) H i G, i; (2) G = H 1, H 2,..., H n = H 1 H 2 H n ; (3) For i j, H i and H j are commutative elementwise, that is, for any h i H i and h j H j, h i h j = h j h i ; (4) H i (H 1 H i 1 H i+1 H n ) = 1, i; (5) Every element h of G can be uniquely expressed as a product of elements in H 1,..., H n : h = h 1 h n, h 1 H 1,..., h n H n.

27 Normal subgroups and direct products 17 Next, we define the inner direct product of subgroups. A group G is called the inner direct product of its subgroups H and K if G = HK and the map (h, k) hk is an isomorphism from the outer direct product H K onto G. For an inner direct product, we use the same notation as an outer direct product, that is G = H K. Similarly, a group G can be defined as the inner direct product of its n subgroups H 1,..., H n. Theorem Let G be a group and let H 1,..., H n be subgroups of G. Then the following are equivalent. (1) The map H 1 H n G defined by (h 1,..., h n ) h 1 h n is an isomorphism. (2) H i G for all i and each h G can be expressed uniquely as a product of elements of H 1,..., H n : h = h 1 h n, h 1 H 1,..., h n H n. (3) H i G for all i, G = H 1,..., H n, and for each i H i j i H j = 1. If any of the conditions (1)-(3) holds, G is called the inner direct product of the subgroups H 1,..., H n. The condition (3) can be weakened to (3 ) H i (H 1 H i 1 ) = 1, i = 2,..., n. In other words, the equations (1), (2) and (3 ) are also a necessary and sufficient condition for G = H 1 H n. Also we have In fact, if G is the inner direct product of its n subgroup H 1,..., H n, it is isomorphic to their outer direct product. Proposition Let G be a group and M G, N G, M N = 1. Then for any m M and n N, we have mn = nm. Proof: Note that m 1 n 1 mn = (m 1 n 1 m)n = m 1 (n 1 mn). Since N G, m 1 n 1 m N; since M G, n 1 mn M. Therefore, m 1 n 1 mn M N. Since M N = 1, we have m 1 n 1 mn = 1, that is mn = nm.

28 18 Basic concepts Proposition Let M G, N G. Then G/(M N) (G/M) (G/N). (The notation H G means that H is isomorphic to a subgroup of G. ) Proof: Consider the map σ : G (G/M) (G/N) defined by g σ = (gm, gn), g G. Clearly, σ is a homomorphism from G to (G/M) (G/N) and Ker σ = M N. By the Fundamental Theorem of Homomorphisms, we have G/(M N) (G/M) (G/N). Let p be a prime number. An element a of a group G is called a p-element if its order o(a) is a power of p. A group G is called a p- group if every element of G is a p-element. Later, it will be shown that a finite group is a p-group if and only if its order is a power of p. (See Exercise ) More generally, for a set π of primes, a group G is called a π-group if every prime divisor of G belongs to π. An element a G is called a π-element if a is π-group. The π-groups will be studied later. Theorem (A classification of finite abelian groups) (1) A finite abelian p-group G can be expressed as a direct product of cyclic subgroups: G = a 1 a s. The number s and the orders p e 1,..., p es of the direct factors (where one may assume that e 1 e s ) are uniquely determined by G. We call (p e 1,..., p es ) the type invariant of G, and call {a 1,..., a s } a basis for G. (2) A finite abelian group G can be expressed as a direct product of cyclic groups: G = a 1 a 2 a s, where o(a i ) o(a i+1 ), i = 1, 2,..., s 1.

Group Theory. Contents

Group Theory. Contents Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

Group Fundamentals. Chapter 1. 1.1 Groups and Subgroups. 1.1.1 Definition

Group Fundamentals. Chapter 1. 1.1 Groups and Subgroups. 1.1.1 Definition Chapter 1 Group Fundamentals 1.1 Groups and Subgroups 1.1.1 Definition A group is a nonempty set G on which there is defined a binary operation (a, b) ab satisfying the following properties. Closure: If

More information

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory

Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

G = G 0 > G 1 > > G k = {e}

G = G 0 > G 1 > > G k = {e} Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same

More information

4. FIRST STEPS IN THE THEORY 4.1. A

4. FIRST STEPS IN THE THEORY 4.1. A 4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We

More information

Elements of Abstract Group Theory

Elements of Abstract Group Theory Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for

More information

GROUPS ACTING ON A SET

GROUPS ACTING ON A SET GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

More information

EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES FOR THE COURSE MATH 570, FALL 2010 EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

More information

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

More information

GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

More information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

More information

2. Let H and K be subgroups of a group G. Show that H K G if and only if H K or K H.

2. Let H and K be subgroups of a group G. Show that H K G if and only if H K or K H. Math 307 Abstract Algebra Sample final examination questions with solutions 1. Suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40, Determine H. Solution. Since gcd(18,

More information

Notes on finite group theory. Peter J. Cameron

Notes on finite group theory. Peter J. Cameron Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geometry (where groups describe in a very detailed way the

More information

ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

Assignment 8: Selected Solutions

Assignment 8: Selected Solutions Section 4.1 Assignment 8: Selected Solutions 1. and 2. Express each permutation as a product of disjoint cycles, and identify their parity. (1) (1,9,2,3)(1,9,6,5)(1,4,8,7)=(1,4,8,7,2,3)(5,9,6), odd; (2)

More information

some algebra prelim solutions

some algebra prelim solutions some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

More information

6 Commutators and the derived series. [x,y] = xyx 1 y 1.

6 Commutators and the derived series. [x,y] = xyx 1 y 1. 6 Commutators and the derived series Definition. Let G be a group, and let x,y G. The commutator of x and y is [x,y] = xyx 1 y 1. Note that [x,y] = e if and only if xy = yx (since x 1 y 1 = (yx) 1 ). Proposition

More information

GROUP ACTIONS KEITH CONRAD

GROUP ACTIONS KEITH CONRAD GROUP ACTIONS KEITH CONRAD 1. Introduction The symmetric groups S n, alternating groups A n, and (for n 3) dihedral groups D n behave, by their very definition, as permutations on certain sets. The groups

More information

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

More information

GENERATING SETS KEITH CONRAD

GENERATING SETS KEITH CONRAD GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

More information

Chapter 13: Basic ring theory

Chapter 13: Basic ring theory Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

More information

ABSTRACT ALGEBRA. Romyar Sharifi

ABSTRACT ALGEBRA. Romyar Sharifi ABSTRACT ALGEBRA Romyar Sharifi Contents Introduction 7 Part 1. A First Course 11 Chapter 1. Set theory 13 1.1. Sets and functions 13 1.2. Relations 15 1.3. Binary operations 19 Chapter 2. Group theory

More information

Introduction to Modern Algebra

Introduction to Modern Algebra Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

More information

Chapter 7. Permutation Groups

Chapter 7. Permutation Groups Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

More information

Group Theory. Chapter 1

Group Theory. Chapter 1 Chapter 1 Group Theory Most lectures on group theory actually start with the definition of what is a group. It may be worth though spending a few lines to mention how mathematicians came up with such a

More information

Chapter 7: Products and quotients

Chapter 7: Products and quotients Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

More information

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

More information

THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP

THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP by I. M. Isaacs Mathematics Department University of Wisconsin 480 Lincoln Dr. Madison, WI 53706 USA E-Mail: isaacs@math.wisc.edu Maria

More information

GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS

GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS GROUPS WITH TWO EXTREME CHARACTER DEGREES AND THEIR NORMAL SUBGROUPS GUSTAVO A. FERNÁNDEZ-ALCOBER AND ALEXANDER MORETÓ Abstract. We study the finite groups G for which the set cd(g) of irreducible complex

More information

Notes on Group Theory

Notes on Group Theory Notes on Group Theory Mark Reeder March 7, 2014 Contents 1 Notation for sets and functions 4 2 Basic group theory 4 2.1 The definition of a group................................. 4 2.2 Group homomorphisms..................................

More information

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute

More information

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

More information

ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

More information

Factoring of Prime Ideals in Extensions

Factoring of Prime Ideals in Extensions Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

COMMUTATIVITY DEGREE, ITS GENERALIZATIONS, AND CLASSIFICATION OF FINITE GROUPS

COMMUTATIVITY DEGREE, ITS GENERALIZATIONS, AND CLASSIFICATION OF FINITE GROUPS COMMUTATIVITY DEGREE, ITS GENERALIZATIONS, AND CLASSIFICATION OF FINITE GROUPS ABSTRACT RAJAT KANTI NATH DEPARTMENT OF MATHEMATICS NORTH-EASTERN HILL UNIVERSITY SHILLONG 793022, INDIA COMMUTATIVITY DEGREE,

More information

Row Ideals and Fibers of Morphisms

Row Ideals and Fibers of Morphisms Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

More information

Some Basic Techniques of Group Theory

Some Basic Techniques of Group Theory Chapter 5 Some Basic Techniques of Group Theory 5.1 Groups Acting on Sets In this chapter we are going to analyze and classify groups, and, if possible, break down complicated groups into simpler components.

More information

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

More information

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B

More information

Universitext. Springer. S. Axler F.W. Gehring K.A. Ribet. Editorial Board (North America): New York Berlin Heidelberg Hong Kong London Milan Paris

Universitext. Springer. S. Axler F.W. Gehring K.A. Ribet. Editorial Board (North America): New York Berlin Heidelberg Hong Kong London Milan Paris Universitext Editorial Board (North America): S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo This page intentionally left blank Hans Kurzweil Bernd

More information

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

More information

(Q, ), (R, ), (C, ), where the star means without 0, (Q +, ), (R +, ), where the plus-sign means just positive numbers, and (U, ),

(Q, ), (R, ), (C, ), where the star means without 0, (Q +, ), (R +, ), where the plus-sign means just positive numbers, and (U, ), 2 Examples of Groups 21 Some infinite abelian groups It is easy to see that the following are infinite abelian groups: Z, +), Q, +), R, +), C, +), where R is the set of real numbers and C is the set of

More information

Galois representations with open image

Galois representations with open image Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group

More information

Chapter 10. Abstract algebra

Chapter 10. Abstract algebra Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

More information

11 Ideals. 11.1 Revisiting Z

11 Ideals. 11.1 Revisiting Z 11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

More information

Classification of Cartan matrices

Classification of Cartan matrices Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

More information

4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2:

4. CLASSES OF RINGS 4.1. Classes of Rings class operator A-closed Example 1: product Example 2: 4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets

More information

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

More information

Group Theory: Basic Concepts

Group Theory: Basic Concepts Group Theory: Basic Concepts Robert B. Griffiths Version of 9 Feb. 2009 References: EDM = Encyclopedic Dictionary of Mathematics, 2d English edition (MIT, 1987) HNG = T. W. Hungerford: Algebra (Springer-Verlag,

More information

1 Sets and Set Notation.

1 Sets and Set Notation. LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

More information

Notes on Algebraic Structures. Peter J. Cameron

Notes on Algebraic Structures. Peter J. Cameron Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester

More information

Geometric Transformations

Geometric Transformations Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

More information

Test1. Due Friday, March 13, 2015.

Test1. Due Friday, March 13, 2015. 1 Abstract Algebra Professor M. Zuker Test1. Due Friday, March 13, 2015. 1. Euclidean algorithm and related. (a) Suppose that a and b are two positive integers and that gcd(a, b) = d. Find all solutions

More information

Abstract Algebra Cheat Sheet

Abstract Algebra Cheat Sheet Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the

More information

Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

More information

Ideal Class Group and Units

Ideal Class Group and Units Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

More information

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

More information

Math 231b Lecture 35. G. Quick

Math 231b Lecture 35. G. Quick Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

More information

FACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set

FACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,

More information

Galois Theory III. 3.1. Splitting fields.

Galois Theory III. 3.1. Splitting fields. Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where

More information

Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

More information

5.1 Commutative rings; Integral Domains

5.1 Commutative rings; Integral Domains 5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

r + s = i + j (q + t)n; 2 rs = ij (qj + ti)n + qtn.

r + s = i + j (q + t)n; 2 rs = ij (qj + ti)n + qtn. Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in

More information

MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

More information

k, then n = p2α 1 1 pα k

k, then n = p2α 1 1 pα k Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

More information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

More information

INTRODUCTORY SET THEORY

INTRODUCTORY SET THEORY M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

More information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

NOTES ON GROUP THEORY

NOTES ON GROUP THEORY NOTES ON GROUP THEORY Abstract. These are the notes prepared for the course MTH 751 to be offered to the PhD students at IIT Kanpur. Contents 1. Binary Structure 2 2. Group Structure 5 3. Group Actions

More information

Week 5: Binary Relations

Week 5: Binary Relations 1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all

More information

ABEL S THEOREM IN PROBLEMS AND SOLUTIONS

ABEL S THEOREM IN PROBLEMS AND SOLUTIONS TeAM YYePG Digitally signed by TeAM YYePG DN: cn=team YYePG, c=us, o=team YYePG, ou=team YYePG, email=yyepg@msn.com Reason: I attest to the accuracy and integrity of this document Date: 2005.01.23 16:28:19

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:

Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold: Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),

More information

Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites

Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites This section consists of an enumeration of terms from elementary set theory and algebra. You are supposed to be familiar with their

More information

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

More information

Basic Algebra (only a draft)

Basic Algebra (only a draft) Basic Algebra (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Kuştepe Şişli Istanbul Turkey anesin@bilgi.edu.tr February 12, 2004 2 Contents I Basic Group Theory 7 1 Definition

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES.

ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES. ON TORI TRIANGULATIONS ASSOCIATED WITH TWO-DIMENSIONAL CONTINUED FRACTIONS OF CUBIC IRRATIONALITIES. O. N. KARPENKOV Introduction. A series of properties for ordinary continued fractions possesses multidimensional

More information

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES 1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

More information

3. Prime and maximal ideals. 3.1. Definitions and Examples.

3. Prime and maximal ideals. 3.1. Definitions and Examples. COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,

More information

THE SIGN OF A PERMUTATION

THE SIGN OF A PERMUTATION THE SIGN OF A PERMUTATION KEITH CONRAD 1. Introduction Throughout this discussion, n 2. Any cycle in S n is a product of transpositions: the identity (1) is (12)(12), and a k-cycle with k 2 can be written

More information

Finite dimensional C -algebras

Finite dimensional C -algebras Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

1 Symmetries of regular polyhedra

1 Symmetries of regular polyhedra 1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an

More information

Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

More information

Introduction to finite fields

Introduction to finite fields Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at

More information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

More information

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

More information