Generating sets, Cayley digraphs. Groups of permutations as universal model. Cayley s theorem

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1 Generating sets, Cayley digraphs. Groups of permutations as universal model. Cayley s theorem Sergei Silvestrov Spring term 2011, Lecture 10 Contents of the lecture Generating sets and Cayley digraphs. Groups of permutations as universal model. Cayley s theorem Typeset by FoilTEX

2 Algebra course FMA190/FMA190F Directed graphs: definition Definition 1. A directed graph (or just digraph) is a finite set of points called vertices and some arcs (with a direction denoted by an arrowhead or without a direction) joining vertices. For each generating set S of a finite group G, we can construct the following Cayley digraph D. The number of vertices in D is G. For any a S, there exist arcs of type a. An arc of type a points from x G to y G if and only if y = xa. If a S and a 2 = e, it is customary to omit the arrowhead from the arc of type a. Example: Cayley digraph for G = Z 6 and S = {1} Example 1. Let G = Z 6 and S = {1}. The Cayley digraph has the form Typeset by FoilTEX 1

3 Example: Cayley digraph for G = Z 6 and S = {2,3} Example 2. Let G = Z 6 and S = {2,3}. Let be an arrow of type 2. Because 3 2 = 0 in Z 6, the arrow of type 3 must be. The Cayley digraph has the form Typeset by FoilTEX 2

4 Algebra course FMA190/FMA190F A characterisation of Cayley digraphs Theorem 1. A digraph G is a Cayley digraph of some generating set H of a finite group G if and only if the following four properties are satisfied. ➀ G is connected. ➁ At most one arc goes from vertex g to a vertex h. ➂ Each vertex g has exactly one arc of each type starting at g, and one of each type ending at g. ➃ If two different sequences of arc types starting from vertex g lead to the same vertex h, then those same sequences of arc types starting from any vertex u will lead to the same vertex v. Cayley used this theorem to construct new groups. For example, the following digraph satisfies all conditions of Theorem 1. If we label by a and group of order 8: e b by b, we obtain a Cayley digraph of a new ab a a 3 b a 2 b a 3 a 2 Typeset by FoilTEX 3

5 Groups of permutations as universal model. Cayley s theorem Permutations and symmetric group Definition 2. A permutation of a non-empty set A is a mapping σ : A A that is both one to one and onto. Let A = {1,2,...,n}. A rearrangement is a list, with no repetitions, of all the elements of A. A rearrangement i 1, i 2,..., i n of A determines a function σ : A A, namely, σ(1) = i 1, σ(2) = i 2,..., σ(n) = i n. We use a two-rowed notation to denote the function corresponding to a rearrangement; if σ( j) is the jth item on the list, then σ = j... n. σ(1) σ(2)... σ( j)... σ(n) That a list contains all the elements of A says that the corresponding function σ is onto; that there are no repetitions on the list says that distinct points have distinct values; that is, σ is one-to-one. Thus, each list determines a one-to-one correspondence σ : A A; that is, each rearrangement determines a permutation. Conversely, every permutation σ determines a rearrangement, namely, the list σ(1), σ(2),..., σ(n) displayed as the bottom row. Therefore, rearrangement and permutation are simply different ways of describing the same thing. The advantage of viewing permutations as functions, however, is that they can now be composed and their composite is also a permutation. Definition 3. The family of all the permutations of a set A, denoted by S A, is called the symmetric group on A. When A = {1,2,...,n}, S A is usually denoted by S n, and it is called the symmetric group on n letters. Typeset by FoilTEX 4

6 Example: S 3 and D 3 Example 3. Denote the elements of S 3 as ρ 0 =, µ 1 =, ρ 1 =, µ =, ρ 2 =, µ 3 = The multiplication table of S 3 is as follows. ρ 0 ρ 1 ρ 2 µ 1 µ 2 µ 3 ρ 0 ρ 0 ρ 1 ρ 2 µ 1 µ 2 µ 3 ρ 1 ρ 1 ρ 2 ρ 0 µ 3 µ 1 µ 2 ρ 2 ρ 2 ρ 0 ρ 1 µ 2 µ 3 µ 1 µ 1 µ 1 µ 2 µ 3 ρ 0 ρ 1 ρ 2 µ 2 µ 2 µ 3 µ 1 ρ 2 ρ 0 ρ 1 µ 3 µ 3 µ 1 µ 2 ρ 1 ρ 2 ρ 0 Consider an equilateral triangle with vertices labelled as 1, 2, and 3. Let ρ j be a rotation by 2π j/3 clockwise, and let µ j be mirror imaging in the bisector of angle j. Then our table is also the multiplication table of the group D 3 of symmetries of an equilateral triangle! Definition 4. The nth dihedral group D n is the group of symmetries of the regular n-gon. The group D 4 is called the octic group. Typeset by FoilTEX 5

7 Cayley s Theorem The next theorem shows that the symmetric groups are incredibly rich and complex. Theorem 2. (Cayley s Theorem, Theorem 7.20, Sec. 7.4, p. 194) Let G be a group. Then G is isomorphic to a subgroup of S G. Sketch of proof. 1. Let f : G S G be the function which sends a G to the function f a : G G defined by f a (g) = ag, g G. For each given a, f a is a one-to-one correspondence between G and itself. 2. f is a homomorphism, i.e., f ab = f a f b. 3. f is one-to-one and thus G is isomorphic to the image f [G] S G. Typeset by FoilTEX 6

8 Orbits Let σ S A. Define the following relation on A: is an equivalence relation. a b n Z: b = σ n (a). Definition 5. Let σ S A. The equivalence classes in A determined by the equivalence relation are the orbits of σ. The algorithm of finding the orbits of σ S n is as follows. Step 1 Pick the smallest element of {1,2,...,n} which has not yet appeared call it a (if you are just starting, a = 1); write {a If no such element exist, stop. Step 2 Read off σ(a) from the given description of σ call it b. If b = a, close the set with a right parenthesis } (without writing b down); this completes an orbit return to Step 1. If b = a, write b next to a in this orbit: {a,b Step 3 Read off σ(b) from the given description of σ call it c. If c = a, close the set with a right parenthesis } to complete the orbit return to Step 1. If c = a, write c next to b in this orbit: {a,b,c Repeat this step using the number c as the new value for b until the orbit closes. Example 4. Let σ S 13 be as follows. ( σ = ). The orbits are {1,12,8,10,4}, {2,13}, {3}, {5,11,7}, {6,9}. Typeset by FoilTEX 7

9 Cycles Definition 6. A permutation σ S n is a cycle if it has at most one orbit containing more than one element. The length of a cycle is the number of elements in its largest orbit. In cyclic notation the cycle is written as σ = (a 1,a 2,...,a m ), where σ(a j ) = a j+m 1, and if a k is not among a 1,, a 2,..., a m, then σ(a k ) = a k. Example 4 shows that σ = (1,12,8,10,4)(2,13)(3)(5,11,7)(6,9), i.e., σ can be written as a product of disjoint cycles. This is a general fact. The cycle decomposition theorem Theorem 3. Every σ S n is a product of disjoint cycles. Proof. Let O 1, O 2,..., O r be the orbits of σ. For 1 j r, define the disjoint cycle µ j as µ j = { σ(x), x O j, x, x / O j. Clearly σ = µ 1 µ 2... µ r. Typeset by FoilTEX 8

10 Transpositions Definition 7. A transposition is a cycle of length 2. Clearly (a 1,a 2,...a m ) = (a 1,a n )(a 1,a n 1 )...(a 1,a 3 )(a 1,a 2 ), i.e., any cycle is a product of transpositions. It follows that if 2 A <, then any σ S A is a product of transpositions. Theorem 4. No permutation in S n can be expressed both as a product of an even number of transposition and as a product of an odd number of transposition. Proof. Let σ S n be a product of j permutations and a product of k permutations. We have to prove that j k is even. Let I n be the n n identity matrix. Because each transposition of rows multiplies the determinant of a matrix by 1, the determinant of σi n is equal to both ( 1) j and ( 1) k. It follows that ( 1) j = ( 1) k, i.e. j k = 2m. The alternating group Definition 8. A permutation σ S n is even (odd) if it can be express as a product of an even (odd) number of permutations. Theorem 5. If n 2, then the set of all even permutations of the set {1,2,...,n} form a subgroup A n of order n!/2 of the symmetric group S n. Definition 9. The group A n is the alternating group on n letters. OBS! The alternating group is a large subgroup of symmetric group. Important Mind puzzle for you before the next lecture! Can you joint to A n some element from S n but not from A n, so that the subgroup generated by this element and A n is not the whole S n? Typeset by FoilTEX 9