Primitive and quasiprimitive groups

Transcription

1 Chapter 4 Primitive and quasiprimitive groups 4.1 Blocks and invariant partitions We start with an example. Let Γ be the circular graph of degree n and D 2n = Aut(Γ) the dihedral group of order 2n. Suppose that the vertex set of Γ is {1,...,n} and the edge set is {{1, 2}, {2, 3},...,{1,n}}. Recall that D 2n is generated by two elements a and b, where n/2 a=(1,2,3,...,n) and b = (i, n i +1) ( n/2 denotes the largest integer that is not greater than n/2). If k is a divisor of n, and m = n/k then the vertex set of Γ can be partitioned into m subset each of size k as follows: i=1 p 1 = {1,m+1,...,(k 1)m +1} p 2 = {2,m+2,...,(k 1)m +2,}. p m = {m, 2m,...,km}. That is α belongs p i if and only if α i mod m. Then it is easy to see that the image of each p i under any element g D 2n is some p j. Thus the permutation group D 2n permutes the parts p 1,...,p m. This motivates the following discussion. 34

2 Groups & Graphs 35 Definition 4.1 Let G be a transitive permutation group on a set Ω and Ω, such that is non-empty. Then is said to be a block for G if for all g G either g = or g =. The following easy lemma shows how we can obtain a partition of Ω from a block. Lemma 4.2 Let G be a transitive permutation group on Ω and Ω ablock.then (i) g G g =Ω; (ii) for all g 1, g 2 G we have that either g 1 = g 2 or g 1 g 2 =. Proof. The proof of (i) is straightforward and is left to the reader. For (ii) suppose that g 1,g 2 Gand ω g 1 g 2.Then ω g 1 1 g 1g 1 1 g 2g 1 1 = g 2g 1 1. As is a block, we obtain that = g 2g 1 1, and hence g 1 = g 2. 2 Exercise 4.3 Let G = D 12 = (1, 2, 3, 4, 5, 6), (1, 6)(2, 5)(3, 4). Find all blocks for G containing the element 1. Exercise 4.4 Let the additive group of the integers Z act on itself by right translation. That is for w, z Z define w z = w + z. Show that if is a block containing 1 then is a subgroup of Z. Exercise 4.5 Let G be a group and consider its right-regular representation. Can you describe all blocks for G containing the identity element? Can you describe all blocks for G? Suppose that G and are as in the previous lemma and let g 1,...,g k be elements of G such that g 1,..., g k is a list of all different images of. That is gi g j if i j and { g g G} = { g 1,..., g k }. Then by the previous lemma Ω= g 1 g k,

3 36 Primitive and quasiprimitive groups where denotes taking disjoint union of two sets. In other words the set P = { g 1,..., g k } is a partition of Ω and P is invariant under the action of G. Such a partition is said to be a G-invariant partition. In other words, each block gives rise to a G-invariant partition of Ω. It is similarly easy to see that an element of a G-invariant partition is a block for G. 4.2 Blocks and subgroups If G is a transitive group of Ω and P is a G-invariant partition of Ω, then G acts transitively on P. This trivial fact is so important that it is worth noting in a lemma. Lemma 4.6 Let G be a transitive permutation group and P a G-invariant partition of Ω. ThenGacts on P and this action is transitive. In particular, each element of P has thesamesize. Proof. Exercise 2 Note that in the previous lemma the G-action on P is, in general, not faithful. The permutation group G P induced by G is in general smaller than G and has smaller degree. Recall that if G is a permutation group on Ω and Γ ΩthenG Γ denotes the setwise stabiliser of Γ, that is G Γ = {g G Γ g =Γ}={g G γ g Γ for all γ Γ}. Lemma 4.7 Let G be transitive permutation group on Ω and ablockforg. Then G is transitive on and G δ G for all δ. Proof. Let δ 1, δ 2. Since G is transitive on Ω, there exists g G, such that δ g 1 = δ 2.Sinceδ 2 g and is a block, we have that = g, therefore g G. This shows that G is transitive on. If δ andg G δ,thenδ=δ g g. As is a block we have that g = and hence g G.ThusG δ G. 2

4 Groups & Graphs 37 Lemma 4.8 Let G be a transitive permutation group acting on Ω and ω Ω afixed element. Let H be a subgroup of G, such that G ω H. Then the orbit ω H of ω under H is a block for G. Proof. Let denote ω H and let g G, andω Ω such that ω g. Then, as ω, there is some h 1 H, such that ω = ω h 1.Sinceω = g we also have that there is some h 2 H, such that ω = ω h2g.thenω h 1g 1 h 1 2 =ω,andsoh 1 g 1 h 1 2 G ω. Set g = h 1 g 1 h 1 2 ;notethatg G ω. Then g = h 1 1 g h 2, and since h 1, h 2 H and g G ω H we have that g H. As is invariant under H,wehave g = as required. 2 Theorem 4.9 If G is a transitive permutation group on Ω and ω is a fixed element of Ω then the correspondence H ω H is a bijection between the set of subgroups containing G ω and the set of blocks containing ω. Moreover for any two such subgroups H 1 and H 2,wehavethatω H 1 ω H 2 if and only if H 1 H 2. Proof. Let ϕ denote the map H ω H. If is a block such that ω, then G is transitive on and G ω G (see Lemma 4.7). Thus = ω G = ϕ(g ), and so ϕ is surjective. Let H 1 and H 2 be subgroups of G, such that G ω H 1 H 2,andω H 1 =ω H 2. Let h 1 H 1. As ω h 1 ω H 1 and ω H 1 = ω H 2 it follows that there exists h 2 H 2,such that ω h 1 = ω h 2.Theng=h 1 h 1 2 G ω,andsoh 1 =gh 2.SinceG ω H 2 we have that h 1 = gh 2 H 2. This shows that H 1 H 2, and similar argument shows that H 2 H 1. Thus H 1 = H 2 and hence ϕ is injective. The proof of the last statement is left as an exercise Primitive and quasiprimitive groups If ω Ω then the singleton {ω} is a block and it gives rise to the partition {{ω} ω Ω} of Ω. Similarly Ω is also a block, and the corresponding partition is {Ω}. These blocks are called trivial blocks. Definition 4.10 A transitive permutation group is said to be primitive if it only has trivial blocks. Otherwise it is said to be imprimitive.

5 38 Primitive and quasiprimitive groups If G is a transitive permutation group and is a non-trivial block then is also called a block of imprimitivity for G. The corresponding G-invariant partition { g g G} of Ω is referred to as a system of imprimitivity for G. In a group G a subgroup H is said to be maximal if H is not properly contained in any proper subgroup of G. In other words, if K G, such that H<Kthen K = G. Corollary 4.11 A transitive permutation group is primitive if and only if a point stabiliser is a maximal subgroup. Proof. This immediately follows from Theorem 4.9. Details are left as an exercise. 2 Exercise 4.12 Recall that a permutation group G on Ω is said to be 2-transitive if for all α 1, α 2, β 1, β 2 Ω, such that α 1 α 2 and β 1 β 2 there exists some g G, such that α g 1 = β 1 and α g 2 = β 2. Prove that any 2-transitive group is primitive. Exercise 4.13 Show that the action of S 5 on the vertices of the Petersen graph is primitive. Theorem 4.14 Let G be a transitive permutation group on Ω and N a normal subgroup of G. Then the orbits of N on Ω form a G-invariant partition of Ω. In particular, each such orbit is a block for G. Proof. We already know that the N -orbits form a partition of Ω, so we only have to prove that this partition is G-invariant. Let be an N -orbit and choose ω. As isann-orbit, we have that = ω N. If g G, then, as N is a normal subgroup, Ng = gn and we obtain that g = {ω ng n N} = {ω gn n N} =(ω g ) N. That is g is the N -orbit containing ω g,andsothesetofn-orbits is G-invariant. We noted earlier that the elements of a G-invariant partition are blocks. 2 Definition 4.15 A permutation group is said to be quasiprimitive if all its non-trivial subgroups are transitive. Corollary 4.16 A primitive permutation group is quasiprimitive.

6 Groups & Graphs 39 Proof. This follows immediately from Theorem Exercise 4.17 Not all quasiprimitive groups are primitive. Can you show an example of a quasiprimitive group that is not primitive. 4.4 Graphs and primitive groups The following important result is due to Charles Sims. Theorem 4.18 (Sims) Let G be a transitive permutation group on Ω. Then G is primitive if and only if each orbital di-graph corresponding to a G-orbital on Ω is connected. Proof. Suppose first that G is primitive and let Γ = (Ω,E) be an orbital di-graph corresponding to the G-orbital E on Ω. Let { 1,..., k } be the partition of Ω to the vertex sets of the connected components of Γ. That is, ω 1, ω 2 i for some i, if and only if there is a path from ω 1 to ω 2. We only have to show that this partition is G-invariant. Let i be a member of this partition, and α i.ifβ g i then there is some α i, such that β = α g. As α i,thereisapathα 0 =α, α 1,...,α k = α from α to α. Taking the image of this path under the element g, we obtain a path from α g to α g = β. Thus g i is contained in the connected component of α g. Similar argument shows that the connected component of α g is contained in g i, therefore g i is the connected component of α g. Hence G permutes the connected components and { 1,..., k } is a G-invariant partition of Ω. As Γ is an orbital graph, each such a connected component has size at least two. The primitivity of G then implies that k =1 and 1 = Ω. Thus Γ is connected. Suppose now that G is transitive on Ω and each orbital graph corresponding to a G-orbital on Ω is connected. Let { 1,..., k } be a G-invariant partition of Ω, such that the size of 1 (and hence that of each i )isatleasttwo. Letα, β 1,and let E =(α, β) G be the orbit of (α, β) under the action of G on Ω Ω. Then E is a G-orbital. Let (γ,δ) E, such that γ 1. Then, as G is transitive on E there is an element g G, such that (α, β) g =(γ,δ), that is α g = γ and β g = δ. Now γ = α g 1 g 1, and, since 1 is a block, we have that g 1 = 1. As β 1, this implies that β g = δ 1. Thus if the initial vertex of an arc is in 1 then so

7 40 Primitive and quasiprimitive groups is its terminal vertex. Similar argument shows that the previous sentence remains true after interchanging the words initial and terminal. Hence 1 contains the vertices of a connected component of Γ. By assumption, Γ is connected and hence 1 =Ωandso G is primitive A glimpse of what s beyond A structure theorem, usually referred to as the O Nan-Scott Theorem, of primitive permutation groups was proved in the 1980 s. Besides M. O Nan and L. Scott, many mathematicians contributed to an appropriate formulation and proof of this result. They include, among others, P. Cameron, L. Kovács, M. Liebeck, J. Saxl, C. Praeger. This theorem divides primitive permutation groups into several classes and gives detailed information about the structure of the group and properties of its action in each class. A similar theorem for quasiprimitive groups was proved by C. Praeger in The discussion of these two theorems is beyond the scope of this course, because it involves constructions, such as wreath product, twisted wreath product. However, in the the following exercises you can taste the flavour of the more involved theory of primitive and quasiprimitive groups. Be warned that these exercises might be a bit more challenging than the previous ones. Let G be a group and N a normal subgroup of G. Then N is said to be a minimal normal subgroup if N does not properly contain any non-trivial normal subgroup of G. In other words, if M is a normal subgroup of G, such that M<N then M =1. Exercise 4.19 Show that if N 1 and N 2 are two minimal normal subgroups of G, then either N 1 = N 2 or N 1 N 2 = 1. Deduce that in the latter case N 1 and N 2 centralise each other. That is n 1 n 2 = n 2 n 1 for all n 1 N 1 and n 2 N 2. Exercise 4.20 Prove that in a group a minimal normal subgroup N is a direct product of isomorphic simple groups. (This might be a bit difficult. However, most group theory textbooks will have such a result; consult one.) In particular, if N is abelian then there is some prime p, such that N = C p C p,wherec p is the cyclic group of order p. Exercise 4.21 Let p be a prime and F p be the field of p elements and V a finitedimensional vector space over F p.let V be the image of the right-regular representation

8 Groups & Graphs 41 of V.Thatis V={ v v V}where for each v V, the permutation v is defined as w v = w + v for all w V. Let GL(V ) denote the group of linear transformations of V. (i) Show that V,GL(V ) = V GL(V )= V GL(V ). The group V GL(V ) is called the affine linear group and is denoted by AGL(n, p). (ii) Show that AGL(n, p) is 2-transitive, and hence primitive, on V. (iii) Show that V is the unique minimal normal subgroup of AGL(n, p). (iv) Let G be a subgroup of GL(V ). Then G is said to be irreducible if {0} and V are the only G-invariant subspaces of G. Show that the following are equivalent: (a) G is irreducible; (b) V is the unique minimal normal subgroup of V G; (c) V G is primitive on V ; (d) V G is quasiprimitive on G. Exercise 4.22 Suppose that G is a quasiprimitive group on Ω and N is an abelian normal subgroup of G. (i) Deduce that N is a minimal normal subgroup G, andn is transitive. Exercise 1.11, show that N is regular. Using, (ii) Show that there is some prime p, such that N = C p C p where C p is the cyclic group of order p (Exercise 4.20). Show that N can be considered as a vector space over F p. (iii) As N is regular we can suppose without loss of generality that Ω = N.ViewNas a vector space and show that N G 0 =1,whereG 0 denotes the point stabiliser of the zero element of N. Using the fact that N is transitive, also show that G = NG 0, and hence G = N G 0. (iv) Viewing N as an F p -vector space, show that G 0 is a group of linear transformations of N ;thatisg 0 GL(N). Show that G 0 is an irreducible group.

9 42 Primitive and quasiprimitive groups Exercise 4.23 Let T be a finite simple group and Aut(T ) its automorphism group. Let T denote the image of the right regular representation of T. (i) Show that T,Aut(T ) = T Aut(T )= T Aut(T ). The group T Aut(T ) is called the holomorph of T and is denoted by Hol T.ThusHol T Sym T. (ii) Prove that Hol T is primitive on T. (iii) Let G be a subgroup of Aut(T ). Then show that the following are equivalent: (a) Inn(T ) G; (b) the only G-invariant subgroups of T are 1 and T ; (c) T G is primitive on T ; (d) T G is quasiprimitive on T.