34 NICK GILL
7. Minimal normal subgroups and the socle Throughout this sectionG is a nontrivial group. A minimal normal subgroup ofG is a normal subgroup K= 1 ofG which does not contain any other nontrivial normal subgroup ofG. The socle, soc(G) is the subgroup� generated by the set of all minimal normal subgroups (ifG has no minimal normal subgroups, then we set soc(G) = 1).
(E7.1)Find the socle ofD 10,A 4,S 4,S 4 Z. × (E7.2)Give an example of a group that has no minimal normal subgroup. (E7.3)IfG is the direct product of a (possibly infinite) number of finite simple groups, then what is soc(G)? (E7.4)Give a characterization of almost simple groups in terms of their socle. Clearly soc(G) is a characteristic subgroup ofG. 26 Theorem 7.1. LetG be a nontrivial finite group. (1) IfK is a minimal normal subgroup ofG, andL is any normal subgroup ofG, then eitherK L or K,L =K L. 27 ≤ (2) There� exist� minimal× normal subgroupsK ,...,K ofG such that soc(G)=K K . 1 m 1 × · · · m (3) Every minimal normal subgroupK ofG is a direct productK=T 1 T k where theT i are simple normal subgroups ofK which are conjugate inG. × · · · (4) If the subgroupsK i in (2) are all nonabelian, thenK 1,...,Km are the only minimal normal sub- groups ofG.
Proof. (1) SinceK L�G, the minimality ofK implies that eitherK L orK L= 1 . In the latter case, since∩ K andL are both normal, we have that K,L =≤KL=K∩L. { } (2) BecauseG is finite we can find a setS= K ,...,K � of� minimal normal× subgroups which is { 1 m} maximal with respect to the property thatH := S =K 1 K m. We must show thatH contains all minimal normal subgroups ofG, and� so� is equal to× soc(· · · ×G). This follows immediately from (1). (3) LetT be a minimal normal subgroup ofK and observe that all conjugatesT g, forg G, are also minimal normal subgroups ofK. Choose a setS= T ,...,T of these conjugates∈ which is { 1 m} maximal with respect to the property thatL := S =T 1 T m. Then, arguing `ala (2), we see thatL contains all of the conjugates ofT under� G� and so×L · ·G ·. × But, since 1 26i.e. soc(G) is invariant under any automorphism ofG. This is because any automorphism ofG must permute the minimal normal subgroups ofG. 27I am considering the internal direct product here, a special case of the internal semidirect product that we have already seen. 28Recall that an elementary-abelianp-group is defined to be a group that is isomorphic toC C , wheren is finite, p × · · · × p n p is a prime, andC p is a cyclic group of orderp. � �� � FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 35 7.1. Finite groups with elementary abelian socle. In this sectionG is a finite group with an elementary-abelian socle. We write V := soc(G)=C C . p × · · · × p d Observe thatV has a natural structure as a vector space� over�� F p�, the field of orderp. This allows us to make the following assertion. Lemma 7.2. G/CG(V) is isomorphic to a subgroup of GL(V), and G/C G(V) acts onV via multiplication (on the right) by matrices. Proof. Consider the action ofGonV by conjugation. Lemma 3.2 implies that this induces a homomorphism φ:G Aut(V ) where we viewV is an object from Group. FurthermoreC G(V) is the kernel ofφ and, in particular,→ it is a normal subgroup ofG. Now, sinceC G(V ) is the kernel of the conjugation action, the first isomorphism theorem of groups implies that G/V is isomorphic to a subgroup of Aut(V ). Now the result follows from (E7.6) below.� (E7.6) LetV be an elementary-abelianp-group. (1) LetG be a group of automorphisms ofV as an object from Group. Prove thatG acts linearly onV , i.e. prove thatG is a group of automorphisms ofV as an object from Vect Fp . (2) LetG be a group of automorphisms ofV as an object from Vect Fp . Prove thatG is a group of automorphisms ofV as an object from Group. (3) Conclude that Aut(V ) = GL(V), whether we consider it an object of Group or of Vect Fp . Let us strengthen our supposition: let us suppose thatG splits overV , i.e. that there exists a subgroup H < G, such thatG=V�H. To state the structure result in this case, we need a definition. Given a vector spaceV over a fieldK, define AGL(V):= (g,v) v K d, g GL(V) =K d � GL(V). { | ∈ ∈ } Here we writeK d to mean the additive group whose elements ared-tuples with entries fromK. Multipli- cation is defined in the usual way for a semidirect product: g2 (g1, v1)(g2, v2) := (g1, g2, v1 v2) n g where, forv K andg GL d(K), we definev to be the product ofv (thought of as a row vector) with g, a matrix.∈ ∈ Note that, just as with GL(V ), we will write AGLd(K) as a pseudonym for AGL(V ). Furthermore, if K=F p is finite, then we will write AGLd(p) as a synonym for AGLd(Fp). Proposition 7.3. Suppose thatG is a finite group with socleV of orderp d for some primep. IfG splits overV , thenG is isomorphic to a subgroup of AGL d(p). Proof. LetH be a subgroup ofG such thatG=V�H. Consider the action ofH onV by conjugation. We claim thatC H (V ) is a normal subgroup ofG; indeed it is enough to prove that it is a normal subgroup ofH, sinceG=VH andC H (V ) is certainly normalized byV. To prove the claim, takeh C H (V),h 1 H andv V . Now observe that ∈ ∈ h ∈ (h 1 ) h1 1 h1 v = (h )− v(h ) 1 1 1 = (h1− hh1)− v(h1− hh1) 1 1 1 =h 1− (h− (h1vh1− )h)h1 1 1 =h 1− (h1vh1− )h1 =v. h1 Thush C H (V ) and the claim is proved. But now, sinceH V= 1 , we know thatC H (V) V= 1 and so, since∈ V is the socle, we conclude thatC (V)= 1 . ∩ { } ∩ { } H { } Now considerC G(V ). Ifg G, theng=vh for a uniquev V andh H andg C G(V) if and only ifh C (V ). In particular,∈ ∈ ∈ ∈ ∈ G CG(V)= V.C H (V)=V. 36 NICK GILL Now Lemma 7.2 implies that G/V is isomorphic to a subgroup of GL(V ). Moreover the groupH ∼= G/V acts onV by right multiplication by matrices; in other wordsV�H is isomorphic to a subgroup of AGL d(p) as required. � (E7.7)Suppose thatK is finite. Prove that soc(AGL (K)) = Kd and, moreover, that soc(AGL (K)) is a minimal normal • d ∼ d subgroup of AGLd(K). Suppose that soc(AGLd(K) G AGL d(K). Under what conditions is soc(AGLd(K))a • minimal normal subgroup of≤G?≤ (E7.8)Describe the structure of AGL1(p) for a primep. 7.2. The socle of a primitive permutation group. Theorem 7.4. IfG is a finite primitive subgroup of Sym(n), andK is a minimal normal subgroup ofG, then exactly one of the following holds: (1) for some primep and some integerd,K is a regular elementary abelian group of orderp d, and soc(G) =K=C G(K). (2)K is a regular non-abelian group,C G(K) is a minimal normal subgroup ofG which is permutation isomorphic toK, and soc(G) =K C G(K). (3)K is non-abelian and soc(G)=K.× Proof. LetC=C G(K). SinceC�G, eitherC=1orC is transitive. SinceK is transitive, Lemma 3.7 implies thatC is semiregular, and hence eitherC=1orC is regular. Suppose thatC = 1. ClearlyK is non-abelian. Furthermore Theorem 7.1 (1) implies thatK is the only minimal normal subgroup ofG and so soc(G)=K and conclusion (3) of the result holds. Suppose instead thatC is regular. Then (E3.23) implies thatC (C) is regular. SinceK Sym(Ω) ≤ CSym(Ω)(C) andK is transitive, we conclude thatK=C Sym(Ω)(C). Similarly,C=C Sym(Ω)(K) and, by (E3.24),C andK are permutation isomorphic. Now, sinceC is regular, we conclude thatC is a minimal normal subgroup ofG (since any proper subgroup ofC is intransitive). Now Theorem 7.1 (1) states that every minimal normal subgroup ofG distinct fromK is contained inC. Thus soc(G) =KC which equalsK orK C depending on whether C K or not. × ≤IfC K, thenC=K,K is abelian, soc(G)=K and conclusion (1) holds. IfC K, then conclusion ≤ �≤ (2) holds. � (E7.9) Suppose thatG is a maximal primitive subgroup of Sym(n). Prove thatG has a unique minimal normal subgroup (and so possibility (2) in Theorem 7.4 cannot occur).29 (E7.10) Suppose thatK is a regular normal subgroup ofG, a subgroup of Sym(n). LetH be the stabilizer of a point in the action onΩ:= 1, . . . , n . ThenG=KH,K H= 1 and, in { } ∩ { } particular,G splits overK, i.e.G=K�H. 7.3. Primitive permutation groups with abelian socle. Our approach now is similar to that of 5 and 6. In those sections we studied subgroupsG of Sym(n) that preserved certain structures on the set§§ 1,...,n (in 5, this was aG-congruence; in 6 it was aG-product structure). In this section the structure of{ interest} is a§G-affine structure. § It will be convenient to take a more categorical approach here, simply becaused the category associated to aG-affine structure is very standard. The category of interest is calledAff K ; it is very similar to VectK , but we are allowing more arrows.30 Objects: Objects are finite-dimensional vector spaces over the fieldK. Arrows: An arrowg:V 1 V 2 is an affine transformation, i.e. a map that acts linearly on the difference between two vectors.→ In other wordsg:V V is an affine transformation if there exists a 1 → 2 29You may find it helpful to refer to the proof of (E3.23) and (E3.24). 30we have done this before – see Example 3 – but in a different direction. FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 37 linear transformationh:V V such that 1 → 2 vg v g = (v v )h. 2 − 1 2 − 1 Suppose thatV is an object in Aff . Forx V define the map K ∈ n :V V,v v+ x. x → �→ 31 The map,n x, is called the translation by the vectorx and one can check thatn x is an arrow in Aff K . The set of all translations N := n x V { x | ∈ } is a subgroup of Aut(V ). It is clear that any linear transformation ofV is also an affine transformation, thus GL(V ) is also a subgroup of Aut(V). (E7.11) LetV be a vector space. Prove that N GL(V)= 1 , whereN is the set of translations ofV; • ∩ { } Aut(V)=N� GL(V) ∼= AGLd(K) whered = dim(V), the dimension ofV as a vector space • overK; AGL(V) acts faithfully and 2-transitively onV; • The stabilizer of the zero vector is GL(V). • Now letG be a subgroup of Sym(Ω) where Ω is a finite set. An affine structure is a bijectionθ:Ω V, whereV is a finite-dimensional vector space over a finite fieldK. An affine structure is aG-affine structure→ 32 ifG acts onV as an object from Aff K . By Lemma 3.2, if aG-affine structure exists, then the action of G on Ω yields a homomorphismG Aut(V ). By (E7.11) Aut(V) ∼= AGLd(p) for some positive integerd and some primep. → Lemma 7.5. LetΩ be a finite set of ordern. Suppose thatθ:Ω V is an affine structure, whereV is → ad-dimensional vector space over a finite fieldK=F p. (1)θ is aG-affine structure for a unique subgroupG of Sym(Ω) that is isomorphic to AGL d(p); (2) ifθisaH-affine structure for some groupH Sym(Ω), thenH G. ≤ ≤ Proof. We useθ to identify Ω withV throughout. We have seen that Aut(V) ∼= AGLd(p) where we consider V an object from Aff K . By (E7.11), the action of AGL(V) onV , as a set, is faithful, thus AGL(V)isa subgroup of Sym(V). Now suppose thatθisaH-affine structure for some groupH Sym(Ω). By definition every element of ≤ h is an arrow in Aff K and so lies in AGL(V ). We are done. � Lemma 7.5 can be combined with Proposition 6.5 to yield the following. Proposition 7.6. LetH Sym(Ω) where Ω < . One of the following holds: ≤ | | ∞ (1)H is intransitive andH Sym(k) Sym(n k) for some1 31Indeed, a translation is precisely an affine map for which the associated linear map is the identity. 32As usual, I am identifying Ω withθ(Ω) so that I can talk of ‘G acting onV ’. If I wanted to make this precise, I would define the action ofG onV via g 1 g v :=θ((θ − (v)) ). 38 NICK GILL (E7.12)Let AGLd(p) = Aut(Ω) forΩ an object from Aff K . Can you specify necessary and sufficient conditions for a subgroupG Aut(Ω) to act primitively onΩ. ≤ The following exercise implies that Proposition 7.6 can be strengthened by requiring that 4 7.4. The socle and affine structures. Our job now is to connect our knowledge about the socle of a primitive permutation group with the work in the previous section on affine structures. Lemma 7.7. Suppose thatG is a subgroup of Sym(Ω) and thatG contains a normal regular subgroupK. LetH be the stabilizer of any point ofω. The action ofG onΩ is permutation isomorphic to the action ofG on the setK given by ϕ:G K K,(g,k ) (k k)h × → 1 �→ 1 whereg=kh for somek K andh H. ∈ ∈ It is important to realise that the given action ofGonK is not an action ofGonK as an object from Group.33 Note, too, that we are not assuming that Ω is finite here. Proof. Fixω Ω and define a functionβ:K Ω, k ω k. SinceK is regular this function is a bijection. Let 1 :G G∈ be the identity map, and let→ψ be the�→ action ofG on Ω given by the embedding ofG in Sym(Ω).→ Now the result is equivalent to proving that the following diagram commutes: ϕ ϕ h G KK (g,k1)(k1k) × (1,βω) βω (1,βω) βω ψ ψ G ΩΩ (g,ωk1 )( ) × ∗ Writeg=kh for somek K andh H. If we follow the diagram from the top-left corner, down and ∈ ∈ across, then ( ) =ω k1g =ω k1kh. On the other hand, if we go right and then down, we obtain ∗ h 1 1 ( )=ω (k1k) =ω h− k1kh = (ωh− )k1kh =ω k1kh ∗ and we are done. � Proposition 7.8. Suppose thatG is a subgroup of Sym(n) and thatG contains a normal regular elementary abelian subgroupV . Then there is aG-affine structureθ:Ω V , and there is a groupM such that G M Sym(n) withM = AGL(V). → ≤ ≤ ∼ Proof. By Lemma 7.7 we know that the action ofG on Ω is permutation isomorphic to the actionϕ onV. Thus it is enough to show thatϕ is an action ofG onV as an object from Aff K . LetH be the stabilizer of 0, the identity element inV , and recall that, by (E7.11),H =GL(V ). Let g G and leth H,v V be the unique elements such thatg=vh. Letv , v V and observe that ∈ ∈ ∈ 1 2 ∈ vg v g = (v +v) h (v +v) h =v h +v v h v h =v h v h = (v v )h 1 − 2 1 − 2 1 h − 2 − 1 − 2 1 − 2 as required. � 33To see why you should think of the effect of this action on the identity ofK. FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 39 7.5. What is left. Recall that we are trying to understand the subgroups of Sym(n), and that, by Proposition 5.8, this amounts to understanding the primitive subgroups. Now Theorem 7.4 gives three possibilities for a primitive subgroup, and Proposition 7.8 gives a full description for possibility (1). If, moreover, we restrict our attention to maximal primitive subgroups of Sym(n), then (E7.9) implies that possibility (2) cannot occur. Thus we can state the following corollary to Theorem 7.4: Corollary 7.9. IfM is a finite maximal primitive subgroup of Sym(n), andK is a minimal normal subgroup ofG, then exactly one of the following holds: d (1)n=p for some primep and some integerd, andM = AGL d(p). (2)K is non-abelian and soc(G)=K. Thus, to understand the maximal subgroups of Sym(n), we need to understand those finite primitive groups that have a unique minimal normal subgroupK that is nonabelian. It is beyond the scope of this course to properly analyse this situation, although we will at least be able to state a theorem pertaining to this situation in the next section. Before we get there, though, let us observe that we have already seen two examples of primitive subgroups of this kind: In Exercise 16 we looked at actions of Sym(n) on the coset spaces of maximal subgroups other than • Alt(n). This situation can be generalized to cover the action of any almost simple groupG on the coset space of a maximal subgroup that does not contain soc(G). In 6 we considered the product action of the wreath productW := Sym(k) Sym(�), and the next • exercise§ shows we have another example. � (E7.14)Letk and� be integers withk 5. Show thatW := Sym(k) Sym(�) has a unique minimal normal subgroup, and give its isomorphism≥ type. �