Sylow Subgroups

Sylow subgroups play a central role in group theory, so constructing them is a natural and important task. However, although there are numerous proofs for the existence of Sylow subgroups, none of them translates easily to an efficient algorithm. Consequently, some of the most intricate algorithmic machinery for permutation groups was first developed in the context of Sylow subgroup constructions.

The theoretical approach originates in [Kantor, 1985b; 1990] where a polynomial-time algorithm is described for the construction of Sylow sub-groups. GivenG ≤ Sym(), first we construct achainG = G1 ≥ G2 ≥

· · · ≥ Gm =1 of normal subgroups with characteristically simple quotients Gi/Gi+1(i.e.,Gi/Gi+1is the product of isomorphic simple groups). Such nor-mal subgroups can be obtained by the methods of Section 6.2 (see especially Section 6.2.7).

For aprime p dividing |G|, we construct aSylow p-subgroup of G by constructing Sylow p-subgroups recursively in G/Gi, for i = 1,2, . . . ,m.

The extension of the solution fromG/Gi toG/Gi+1requires algorithms for the following two problems:

(1) Find aSylowp-subgroup inGi/Gi+1.

(2) Given two Sylow p-subgroups ofGi/Gi+1, find an element ofGi/Gi+1

conjugating one to the other.

These problems are trivial ifGi/Gi+1 is elementary abelian, but ifGi/Gi+1

is nonsolvable then their solution requires some form of constructive recog-nition of the simple groups in the direct productGi/Gi+1. For the definition of constructive recognition, we refer to Section 8.3. Constructive recognition identifies the simple groups under consideration with a standard copy. For clas-sical groups of Lie type, the standard copy is a group of matrices with “nice”

generating sets, and problems (1) and (2) can be solved by linear algebra. How-ever, for exceptional groups of Lie type, the current solutions of (1) and (2) are essentially by brute force. The running time is polynomial in the input length, but the algorithm is impractical. An important direction of future research is the efficient algorithmic handling of exceptional groups of Lie type, including the solutions of (1) and (2).

Later theoretical developments, namely the nearly linear-time Sylow sub-group computations of [Morje, 1995; 1997] in sub-groups with no exceptional composition factors and the parallel machinery of [Mark, 1993] and [Kantor et al., 1999], all follow this pattern. In this section we shall only describe how to construct aSylow subgroup ofG/Gi+1 in the case when Gi/Gi+1 is ele-mentary abelian. In particular, solving this special problem is enough for the construction of Sylow subgroups in solvable groups.

The first practical algorithm in [Cannon, 1971; Butler and Cannon, 1989]

constructs aSylowp-subgroup by finding larger and largerp-subgroups, based on Exercise 7.3. If a p-subgroupH≤G is already constructed but H is not a Sylow subgroup then the algorithm constructs the centralizers CG(h) for elements of Z(H) until someh is found with the p-part ofCG(h) exceeding

|H|. Then, recursively, aSylow subgroup ofCG(h) is constructed. In the group

CG(h), we can factor out a normalp-group using Theorem 6.3.1 and work in a smaller quotient group. This algorithm uses centralizer computations and thus backtrack searches, which we shall discuss in Chapter 9.

Later practical versions gradually converged toward the theoretical approach just described. In [Atkinson and Neumann, 1990], [Butler and Cannon, 1991], and [Cannon et al., 1997], the method of [Butler and Cannon, 1989] is combined with a divide-and-conquer approach that utilizes block homomorphisms and transitive constituent homomorphisms. In the most recent implementations, a Sylowp-subgroup ofG/O_∞(G) is lifted to aSylow p-subgroup ofGthrough the elementary abelian layers ofO_∞(G), as in the theoretical solution.

After this lengthy introduction, we can now get down to work. We would like to solve the following problem:

GivenQG≤Sym(),with|| =n, such thatQis elementary abelian andG/Qis a p-group, construct aSylow p-subgroup ofG

in nearly linear time by a deterministic algorithm.

(7.2) The approach we present for solving (7.2) is a straightforward sequential modi-fication of the parallel procedure in [Kantor et al., 1999]. Although we describe a deterministic version, randomized methods are very well suited to speed up Sylow subgroup computations. If we make sure that the orders of the input group and of the output (the alleged Sylow subgroup) are computed correctly, then we can check whether the output is correct.

LetQ ∼=C_q^mfor some primeq. Ifq = pthenGis a p-group and we are done. Hence we assume now thatq =p. We compute an SGS{z1, . . . ,zm}for Qthat supports the coordinatization of the elements ofQ, as described in the introduction of Section 7.3.

First, we solve (7.2) in the special case whenG/Q∼=C^r_p is an elementary abelian p-group. Note that in this case any Sylow p-subgroupP ofGis ele-mentary abelian. Forx∈G, we denote by ¯xthe cosetQxinG/Q.

Lemma 7.3.1. Let QG≤Sym()be given, with Q∼=C_q^mand G/Q∼=C^r_p for two different primes p,q. Then a Sylow p-subgroup of G can be found in nearly linear time by a deterministic algorithm.

Proof. Let{z₁, . . . ,zm}be the SGS ofQsupporting coordinatization. First, we constructx1, . . . ,xr ∈Gsuch that ¯x1, . . . ,x¯rgenerateG/Q. This can be done in the faithful permutation representation ofG/Qprovided by Theorem 6.3.1, or just by picking asubsequence of the given generators of G that define a

subgroup chain with groups having orders with increasing p-parts. After that, we compute the matricesT₁, . . . ,Trof the linear transformations corresponding to the conjugation action ofx1, . . . ,xr onQ.

It is enough to findq₁, . . . ,qr ∈Qsuch thatxiqiandxjqjcommute (inG) for alli,j ∈[1,r]. Having accomplished that, theqth powers of the permutations xiqi generate a Sylowp-subgroup ofG.

Sincez1, . . . ,zmis a basis of the vector space Q, theqk can be written in the formqk =m

l=1αklzl, where theαkl are (a priori unknown) elements of GF(q). For fixedi,j ∈[1,r], the condition [xiqi,xjqj]=1 providesmlinear equations for theαkl, since

1=[xiqi,xjqj]=q_i⁻¹x_i⁻¹q⁻¹_j x⁻¹_j xiqixjqj

=q_i⁻¹· q⁻_j¹x⁻¹_i

·[xi,xj]·q_i^xj ·qj. (7.3) All five terms on the second line of (7.3) are elements ofQ. We can compute the coordinate vectors of these elements (as functions of theαkl) since [xi,xj] is a known element ofQ, and conjugation byxi andxj are linear transformations of Q with known matrices Ti andTj, respectively. Hence, for example, the coordinate vector ofq_i^xj is the product of the coordinate vector ofqi and the matrixTj. The sum of the five coordinate vectors corresponding to the terms of the second line of (7.3) is 0 and each coordinate provides a linear equation for theαkl. Altogether, we get (^r₂)m∈O(log³|G|) equations inmr ∈O(log²|G|) unknowns. This system of equations has a solution (since Sylow subgroups exist), and we can find a solution by Gaussian elimination, usingO(log⁷|G|)

field operations in GF(q).

IfG/Qis not elementary abelian then we use an algorithmic form of the

“Frattini argument” (cf. Exercise 7.4). First, we compute a homomorphism ϕ : G → Sn with kernel Q using Theorem 6.3.1, and then we compute an elementary abelian normal subgroup ¯A ∼= C^r_p of ϕ(G) and its preim-age A := ϕ⁻¹( ¯A). By Lemma7.3.1, we can compute aSylow p-subgroup P = x₁, . . . ,xrof A. As a final preparatory step, we set up the data struc-ture (cf. Section 5.1.2) for computing with the isomorphismψ:P= x1, . . . , xr →A¯ = ϕ(x1), . . . , ϕ(xr), which is the restriction ofϕ to P. The point is that, given ¯x ∈ A, we can compute¯ x ∈ P withϕ(x) = x. As before, let¯ {z1, . . . ,zm}be the SGS forQ∼=C_q^msupporting coordinatization.

Now comes the Frattini argument. SinceNG(P)A/A∼=G/A, the normalizer NG(P) contains a Sylow p-subgroup of G. Our next goal is to construct generators for NG(P), by computing some yg∈Afor each generatorg ofG such thatgygnormalizesP. In fact, sinceA=P Q, we shall constructyginQ.

Letgbe afixed generator ofG. SinceAG, we ha vex^g_j ∈ Afor 1≤ j ≤r and so there existx_j ∈ P andqj ∈ Qsuch thatx^g_j = x_jqj. In fact,x_j and qj can be constructed, sincex_j =ψ⁻¹(ϕ(xj)^ϕ(g)) a ndqj =x^g_j/x_j. Hence we need yg ∈ Qsuch that (x_jqj)^yg = x_j for 1 ≤ j ≤ r. This last equation is equivalent to

y⁻¹_g xj·yg·qj =1, (7.4)

which, as in the proof of Lemma 7.3.1, leads to a system of linear equations for the coordinates ofygin the basis{z₁, . . . ,zm}. The system of equations we have to solve consists ofmr ∈O(log²|G|) equations.

GivenG= T, we computeygfor eachg ∈T. Then the set{gyg|g ∈T} generatesNG(P). Next, using Theorem 6.3.1, we construct an action ofNG(P) withPin the kernel; repeating the foregoing procedure in the image, eventually we can discard the p-part ofNG(P) and so obtain a Sylowp-subgroup ofG.

Note that the number of recursive calls in the procedure is bounded by the maximal length of subgroup chains inG.

We can also solve the conjugation problem for Sylow subgroups in solvable groups.

Lemma 7.3.2. Suppose that QG≤Sym(),|| =n,Q is an elementary abelian q-group, and G/Q is a p-group for distinct primes p and q. Given Sylow p-subgroups P₁,P₂in G, we can construct an element y∈ G conjugating P₁ to P2in nearly linear time by a deterministic algorithm.

Proof. We start by constructing an SGS{z1, . . . ,zm}forQsupporting coordina-tization and constructing the homomorphismϕ :G→Snwith kernelQ, as de-scribed in Theorem 6.3.1. We also construct the isomorphismsψ1: P1→ϕ(G) andψ2: P2→ϕ(G), which are the restrictions ofϕtoP1andP2, respectively.

The mapsψi define an isomorphism betweenPiandG/Q, fori =1,2.

SinceP1Q=G, there existsy∈QwithP₁^y=P2, and our aim is to construct aconjugating element inQ. For anyg∈ P₁andy∈ Q, the conjugateg^yis in the cosetQg; therefore, ifyconjugatesP1toP2theng^y=ψ₂⁻¹(ψ1(g)) for all g∈ P₁.

Based on the observations in the previous paragraph, it is straightforward to write a system of linear equations for the coefficients of a conjugating element yin the basis{z₁, . . . ,zm}. Let P₁ = T. The conditiong^y =ψ₂⁻¹(ψ1(g)) is equivalent to

(y⁻¹)^g·y=g⁻¹ψ₂⁻¹(ψ1(g)). (7.5)

Writing (7.5) for allg ∈ T and noticing thatg⁻¹ψ₂⁻¹(ψ1(g))∈ Q, we obtain

|T|mequations for the coefficients ofy.

Theorem 7.3.3. Given a solvable group H≤Sym(),with|| =n, and two Sylow p-subgroups P1,P2of H , an element of H conjugating P1to P2can be constructed by a deterministic nearly linear-time algorithm.

Proof. We start with the computation of a series of normal subgroups 1 = KrKr−1· · · K1 = H with elementary abelian quotients Ki/Ki+1 and homomorphismsϕi:H→Snwith kernelKias in Section 6.3.1, at the construc-tion ofO_∞(H). Then, recursively fori = 1, . . . ,r, we constructgi ∈ ϕi(H) such thatϕi(P₁)^gi=ϕi(P₂). We can start withg₁:=1. Suppose thatgi is al-ready constructed for somei, and let hi∈H be an arbitrary preimage ofgi

underϕ_i⁻¹. IfKi/Ki+1is a p-group thengi+1:=ϕi+1(hi) conjugatesϕi+1(P₁) to ϕi+1(P2). If Ki/Ki+1 is a q-group for some prime q =p then in the group G:= ϕi+1(Ki), ϕi+1(P2), the subgroup Q:=ϕi+1(Ki) is an elemen-tary abelian normalq-subgroup withG/Qap-group, andϕi+1(P₁^hi), ϕi+1(P₂) are two Sylowp-subgroups ofG. Hence we are in the situation of Lemma 7.3.2 and we can constructy∈Gconjugatingϕi+1(P₁^hi) toϕi+1(P₂). Then we define gi+1:=ϕi+1(hi)y. At the end of the recursion,gr ∈ϕr(H)∼=His the element

ofHconjugating P₁toP₂.