A Library of Nearly Linear-Time Algorithms
Theorem 6.1.13. The algorithm Centralizer of Normal Subgroup in Tran- Tran-sitive Group(G, N) is deterministic and it runs in nearly linear time
6.3. Quotients with Small Permutation Degree
6.3.1. Solvable Radical and p-Core
For agroupG, the solvable radical(in short, radical)O∞(G) is defined as the largest solvable normal subgroup ofGand, for prime numbers p, the p-core Op(G) is the largest normal p-group inG. Note that, since two solvable nor-mal subgroups generate a solvable nornor-mal subgroup, there is a unique maxinor-mal solvable normal subgroup in G. Similarly, G has a unique maximal normal p-subgroup. By the same argument, we see thatO∞(G) a ndOp(G) are char-acteristic inG.
Based on Theorem 6.3.1, [Luks, 1990] gave polynomial-time algorithms for computingO∞(G) a ndOp(G). Forp-cores, another polynomial-time algorithm is described in [Neumann, 1986], and p-cores can be also obtained as the intersection of Sylow p-subgroups (cf. [Butler and Cannon, 1989]), although the computation of a Sylow subgroup is much more difficult than obtaining the p-core. Here we follow the nearly linear-time version from [Luks and Seress, 1997]. The basic idea is the same as in [Luks, 1990]. The radical andp-core are computed similarly, so we handle both of them simultaneously. Our primary
description is the computation of the radical; the necessary changes for the computation of thep-core are indicated in double brackets.
First, let us consider the special case whenGhas a maximal normal subgroup NwithO∞(N)=1[[Op(N)=1]]. It turns out that this extra condition restricts severely the radical andp-core ofG.
Lemma 6.3.2. Suppose that N≤G is a maximal normal subgroup and O∞(N)=1[[Op(N)=1]]. Then, if G/N is not cyclic[[not a p-cycle]]then O∞(G)=1[[Op(G)=1]].
Proof. Since the radical andp-core ofGintersectNtrivially, they must central-izeN. In particular, if the radical [[p-core]] ofGis nontrivial thenCG(N) ha s nontrivial radical [[p-core]]. IfCG(N)≤N then its radical [[p-core]] is triv-ial. IfCG(N) ≤N thenG = N CG(N) a ndCG(N)/Z(N)∼=G/N, soCG(N) has a chance to have nontrivial radical [[p-core]] only ifG/N is cyclic [[is a
p-cycle]].
Lemma 6.3.3.
(i) Suppose that N≤G is a maximal normal subgroup, O∞(N)=1, and G/N is cyclic. Then O∞(G) =1if and only if CG(N) =1.
(ii) Suppose that N≤G is a maximal normal subgroup, Op(N)=1, and G/N is a p-cycle. Then Op(G) =1if and only if CG(N) ≤ N .
Proof. (i) SinceZ(N)=1, we must haveCG(N)=1 orCG(N)∼=G/N. In the latter case,O∞(G)=CG(N) =1.
(ii) First, we observe thatOp(Z(N))=1, sinceOp(Z(N))N. IfCG(N)≤ N thenCG(N)=Z(N) and so Op(CG(N))=1. IfCG(N) ≤N thenCG(N)/
Z(N)∼=CpandCG(N)= Z(N),gfor anyg∈CG(N)\Z(N). Sinceg com-mutes with N, it commutes with Z(N). ThereforeCG(N) is abelian, which impliesOp(CG(N))∼=CpandOp(G)=Op(CG(N)) =1.
The Algorithms
Now we are ready to describe the algorithms. First, we compute a composition series 1= N1 N2 · · · Nm =G. By the results of Section 6.2, this can be done in nearly linear time. Then we find the smallest index i such that Ni has a nontrivial radical [[p-core]] by the following method: Suppose that O∞(Ni)=1[[Op(Ni)=1]]. IfNi+1/Niis not cyclic [[not ap-cycle]] then, by Lemma6.3.2, we conclude thatO∞(Ni+1)=1[[Op(Ni+1)=1]]. Otherwise, we computeCNi+1(Ni), using the algorithm of Section 6.1.4. IfCNi+1(Ni) ≤Ni
then, by Lemma 6.3.3, we get a nontrivial radical [[p-core]]. We take its normal closureHinG; by Exercise 6.17(ii), it is a solvable normal subgroup [[normal p-group]] inG.
Next, we compute the derived series of H. The last nontrivial term in the derived series is an abelian normal subgroup [[abelian normalp-group]]N of G. We compute the homomorphismϕ :G → Sn described in Theorem 6.3.1 and repeat the procedure in the image ofϕ. Note that the composition series computation need not be repeated, since 1=ϕ(N1)ϕ(N2)· · · ϕ(Nm)= ϕ(G), withϕ(Ni+1)/ϕ(Ni)∼=Ni+1/Ni orϕ(Ni+1)/ϕ(Ni)=1.
Theorem 6.3.1 and computing O∞(G) are becoming more and more important. In any groupG, we can define a series of characteristic subgroups 1≤N1≤N2≤N3≤G, where N1=O∞(G),N2/N1=Soc(G/N1)∼=T1×
· · · × Tm is the direct product of nonabelian simple groups, N3/N2 Out(T1)× · · · ×Out(Tm), and G/N3 is apermutation group of degree m, corresponding to the conjugation action ofG on{T1, . . . ,Tm}. Constructing these subgroups is one of the main approaches for matrix group computations (cf. [Babai and Beals, 1999; Kantor and Seress, 2002]), but a number of recent permutation group algorithms utilize them as well, for example, for computing conjugacy classes (cf. [Cannon and Souvignier, 1997; Hulpke, 2000]), maximal subgroups (cf. [Eick and Hulpke, 2001; Cannon and Holt, 2002]), or automorphism groups (cf. [Holt, 2001]).
In the permutation group setting, during the construction ofO∞(G) we obtain faithful permutation representations for the factor groupsG/Kifor asequence of normal subgroups 1=Kr≤Kr−1≤ · · · ≤K1=N1=O∞(G), with elemen-tary abelian factorsKi/Ki+1. First we solve the problem at hand (for example, the computation of conjugacy classes) forG/O∞(G). The bulk of the work in this step is the handling of the simple nonabelian groupsTjin Soc(G/O∞(G)), which can be done by identifying these groups with a standard copy of theTj
(cf. Section 8.3 for details and a precise formulation of the identification). In the standard copy, the problems indicated here can be solved more easily than in arbitrary permutation representations. After that, we lift the result recursively fromG/KitoG/Ki+1, fori ∈[1,r−1]. For that task, techniques for solvable permutation groups are used. We shall give examples of such lifting procedures in Section 7.3.
Exercises
6.1. LetG≤Sym() be primitive and 1 =NG. Prove thatNis transitive.
6.2. Finish the proof of Lemma6.1.9.
6.3. Construct apermutation group G≤Sym(), with || =n, and with (n/2√logn) orbits that are of the same size but pairwise not equivalent
in the equivalence relation defined in Section 6.1.2.Hint:We may choose Gto be an elementary abelian 2-group.
6.4. LetH,G≤Sym() such thatGnormalizesH. Prove thatGnormalizes CSym()(H).
6.5. Prove that ifH≤GthenHGif and only if the subgroup chain defined recursively asH0:=G,Hi := HHi−1fori >0 reachesH.
6.6. [Neumann, 1986] Let = {α1, . . . , α4m}. Fori ∈ [1,m], letGi be a dihedral group of order 8, acting on{α4i−3, α4i−2, α4i−1, α4i}and fixing the other 4m−4 points of. Letzi be the generator of Z(Gi). Prove that the subgroupN := z1z2,z1z3, . . . ,z1zmis normal inG:=G1×
· · · ×GmandG/N has no faithful permutation representation of degree less than 2m+1.
6.7. LetG≤Sym() be primitive, with two regular normal subgroups N1
andN2. Prove thatN1∼=N2∼=N for some characteristically simple non-abelian groupNand thatcan be identified withNsuch thatN1andN2
are the right-regular and left-regular representations ofN on itself.
6.8. Let 1 = K≤Sym() and 1 = H≤Sm. We define an action of K H on the set m by the rule that if g := (k1, . . . ,km;h) ∈ K H and δ :=(δ1, . . . , δm)∈mthenδg :=(δ1k1¯h¯h, . . . , δmkm¯h¯h) for ¯h :=h−1. (This is called theproduct actionof the wreath product.)
Prove that this product action is primitive if and only ifK acts primi-tively but not regularly onandHis atransitive subgroup ofSk. 6.9. LetT1, . . . ,Trbe nonabelian simple groups and letHbe asubdirect
prod-uct of the Ti,i =1,2, . . . ,r. Prove that there is a partition (1, . . . , l) of{1,2, . . . ,r}such that, for each fixed j ∈ [1,l], all groups Tk with k∈jare isomorphic, and there is a diagonal subgroupDj≤
k∈jTk
such thatH =l j=1Dj.
6.10. LetT1, . . . ,Trbe nonabelian simple groups. Prove that ifMis aminimal normal subgroup ofT1× · · · ×TrthenM =Ti for somei ∈[1,r].
6.11. For any groupG, prove that the following are equivalent:
(i) Ghas a faithful transitive permutation representation that is not regu-lar, but the stabilizer of any two points is trivial (i.e.,Gis aFrobenius group).
(ii) Ghas a nontrivial subgroup H such that NG(H)=H and any two conjugates ofH are identical or their intersection is 1 (such a sub-groupHis called aFrobenius complement).
6.12. This is not an exercise but rather a compilation of results about Frobe-nius groups. Proofs can be found, for example, in Sections 17 and 18 of [Passman, 1968]. LetG≤Sym() be aFrobenius group and letα∈. ThenH:=Gαis aFrobenius complement forG.
(i) (Frobenius) The set{1} ∪(G\ g∈GHg) is a regular normal sub-group ofG, which is called theFrobenius kernel.
(ii) (Thompson) The Frobenius kernel is nilpotent.
(iii) If|H| is even then H has a unique element of order 2, which is therefore central.
(iv) (Zassenhaus) If|H|is odd then for every prime divisorpof|H:H|, elements of orderpinHare central.
(v) (Zassenhaus) IfHis nonsolvable thenHhas a normal subgroup H0 of index 1 or 2 such thatH0 ∼=SL2(5)×Sfor some solvable group Sof order relative prime to 30.
6.13. Prove Lemma6.2.12.Hint:Use induction onm.
6.14. Letpbe aprime number.
(i) LetPbe the set ofd×dlower triangular matrices over GF(p), with all diagonal entries equal to 1. Prove thatP is aSylowp-subgroup of GLd(p).
(ii) ForH≤GLd(p), let fix(H) denote that the set of vectors in GF(p)d fixed by each element ofH. Prove that ifHis ap-group then fix(H) is anonzero subspace of GF(p)d.
(iii) Prove that if H≤GLd(p) then fix(Op(H)) is an H-invariant sub-space.
6.15. LetG≤Sym() be primitive with two regular normal subgroups and let α∈. Prove that the only minimal normal subgroup ofGαis Soc(G)α. 6.16. Design a faster version of the algorithm described in Lemma 6.2.15, where the intermediate groupshiHfor 1≤i<lare not computed; in-stead, guarantee that a subgroup ofhiHof order greater thannis con-structed with sufficiently high probability and thathi+1is chosen from this subgroup.
6.17. Suppose thatTGandT is simple. Prove that
(i) if T is nonabelian then TG is adirect product of simple groups isomorphic toT and it is a minimal normal subgroup ofG;
(ii) ifT is cyclic of orderpthenTGis a p-group;
(iii) it is possible thatTGis not elementary abelian in part (ii).
6.18. Combining ideas from Sections 6.2.5 and 6.2.6, design a fast Las Vegas algorithm to decide whether a primitive group of degree at most 107and of known order is simple.