Strong Generators in Solvable Groups

Solvable Permutation Groups

7.1. Strong Generators in Solvable Groups

Exploiting special properties of solvable groups, [Sims, 1990] describes a method for constructing a strong generating set. Recall that a finite groupGis solvable if and only if it ispolycyclic, that is, there exists a sequence of elements (y₁, . . . ,yr) such thatG= y1, . . . ,yrand for alli ∈[1,r−1],yinormalizes Hi+1 := yi+1, . . . ,yr.

The main idea is that given an SGS for a groupH ≤Sym() a ndy∈Sym() such thatynormalizesH, an SGS forH,ycan be constructed without sifting of Schreier generators. The method is based on the following observation.

Lemma 7.1.1. Suppose that G= H,y ≤Sym()and y normalizes H . For a fixedα∈, let=α^H, =α^G, and m= ||/||. Then

(i) m is an integer and there exists h∈ H such that z:=y^mh fixesα;

(ii) z normalizes H_α;and (iii) G_α= H_α,z.

Proof. (i) By Lemma6.1.7,is the disjoint union ofG-images of. Moreover, theG-images ofare cyclically permuted byyandmis the smallest integer such that^ym =. In particular,α^ym∈, so there existsh∈ Hwith the desired property.

(ii) Sincey^mandh normalize H, so does their productz. Moreover, since z∈G_α, it normalizesH_α=G_α∩H.

(iii) Anyx ∈Gcan be written in the formx =hxyⁱ for somehx ∈ H and some integeri ≥0. If, in addition, x ∈G_α then^x =and somdividesi.

Thereforex =hx(zh⁻¹)^i/m =uz^i/mfor someu ∈ H. Since bothxandzfix α, so doesu. This implies thatx∈ Hα,z.

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Lemma 7.1.2. Suppose that G = H,y ≤Sym(),with|| = n, and y normalizes H . Moreover, suppose that a nonredundant base B and an SGS S relative to B is given for H , and let t denote the sum of depths of the Schreier trees defined by S. Then there is a deterministic algorithm that computes an SGS S^∗⊇S for G in O(n(t+log|G:H|))time, satisfying|S^∗| ≤ |S| +2 log|G: H| and that the sum of depths of the Schreier trees defined by S^∗ is at most t+2 log|G:H|.

Proof. Let B =(β1, . . . , βM) be the nonredundant base ofH, and letm1 :=

|β₁^G|/|β₁^H|. By Lemma7.1.1(i),m1is an integer. First, we determine the value ofm₁. This is done by computing the imagesβ₁^yi for positive integersi. The smallesti such thatβ₁^yi is in the fundamental orbit:=β₁^Hgives the value of m1. After that, we compute the permutationsy²ⁱ for allisatisfying 2ⁱ ≤m1.

The Schreier tree (S₁^∗,T₁^∗) forGmodG_β1is constructed by augmenting the Schreier tree (S1,T1) forHmodH_β1, using the labelsy²ⁱ. Lett1denote the depth of (S1,T1). Then points in^yj are of distance at mostt1+b(j) from the root in (S₁^∗,T₁^∗), whereb(j) denotes the number of 1s in the binary expansion of j. Hence the depth of (S₁^∗,T₁^∗) is at mostt1+ logm1 ≤t1+2 logm1. We finish processing the first level by computingy^m1and the coset representative h defined by (S1,T1) such that y^m1h fixes β1. By Lemma7.1.1(iii) we have G_β1= H_β1,y^m1h.

The time requirement isO(n) for computingm1,O(nlogm1) for computing they²ⁱ for 0<i ≤logm1,O(nlogm1) for computing (S₁^∗,T₁^∗), andO(n(t1+ logm1)) for computingy^m1h.

After processing the first level, we apply recursively the same procedure for G_{(β1,...,βj)}, for j=1,2, . . . ,M. Denoting the depth of the jth Schreier tree for Hbytjand|β^G(β1,...,βj−1 )

j |/|β^H(β1,...,βj−1 )

j |bymj, the total time requirement of the algorithm isO(nM

j=1(tj+logmj))=O(n(t+log|G: H|)).

Following Sims, we call the algorithm described in the proof of Lemma 7.1.2 Normalizing Generator.

Corollary 7.1.3. Given a polycyclic generating sequence(y1, . . . ,yr)for a solvable group G≤Sn, an SGS supporting membership testing in O(nlog|G|) time per test can be computed in O(r nlog|G|)time by a deterministic algo-rithm. The memory requirement is O(nlog|G| +r n).

In particular, ifG= Tis abelian then the given generators form a polycyclic generating sequence and an SGS can be computed inO(|T|nlog|G|) time by adeterministic algorithm.

Unfortunately, the input generating set for a solvable group is frequently not apolycyclic one and often we do not even know whether the input group is solvable. Hence we useNormalizing Generatoras a subroutine of the follow-inggeneralized normal closure(GNC) algorithm. The input ofGNCis an SGS for someNGandy∈ Gsuch thaty ∈ N. The output is either an SGS for the normal closureM = N,y^Gor two conjugatesu, vof ysuch that the commutatorw=[u, v] ∈N.

Starting withN =1, repeated applications ofGNCconstruct an SGS for a solvable group. If we get an output of the first type, the next call is made withM and a generator ofGnot inM. When an output of the second type is returned, the next call is made with the sameN as before and withw. In this case, the progress we made is thatwis in the derived subgroupG. All conjugates ofw remain inG, so anew output of the second type will be inG. Clearly, in a solvable group sooner or later an output of the first type occurs.

If the input group is not solvable then recursive calls ofGNCcontinue in-definitely. To handle this possibility, a result from [Dixon, 1968] is used. Dixon showed that the derived length of a solvable permutation group of degreen is at most (5 log₃n)/2. Hence, if more than (5 log₃n)/2 consecutive outputs of GNC are of the second type then we stop and conclude that G is not solvable.

The Generalized Normal Closure Algorithm

To finish our description of the solvable group SGS construction, we indicate how the procedureGNCworks. Given N andy, we start a standard normal closure algorithm (cf. Section 2.1.2), collecting the conjugates of generators in asetU. We add only those conjugates that increaseN,U. Each time a permutationuis added toU, we check whether the commutators ofuwith all previously defined elements ofU are inN. If some commutator is not in N then we have found an output of the second type. If all commutators are inN then in particularu normalizesN,U\{u}andNormalizing Generatorcan be used for constructing an SGS forN,U.

We analyze the version of the algorithm when the inverses of transversals are stored in Schreier trees using the method of Lemma 7.1.2. LetT denote the set of given generators forG. The procedureGNCis called O(log|G|logn) times, because outputs of the first type define an increasing chain of subgroups that must have lengthO(log|G|), and between two outputs of the first type only O(logn) outputs of the second type occur.

Within one call ofGNC, the setUincreasesO(log|G|) times since the groups N,Udefine an increasing subgroup chain. HenceO(log²|G|) commutators and O(|T|log|G|) conjugates of the elements of U must be computed and

their membership must be tested in the groups N andN,U, respectively.

By Corollary 7.1.3, one membership test costsO(nlog|G|). By Lemma7.1.2, the total cost ofO(log|G|) calls ofNormalizing GeneratorisO(nlog²|G|).

Combining these estimates, we obtain the following theorem.

Theorem 7.1.4. An SGS for a solvable permutation group G = T ≤ Sn

can be computed in O(nlognlog³|G|(|T| +log|G|))time by a deterministic algorithm.

The solvable SGS construction belongs to the rare species ofdeterministic nearly linear-time algorithms. An additional bonus is that we also obtain a polycyclic generating sequence (and so a composition series). Moreover, in practice, the algorithm runs faster than other speedups of the Schreier–Sims procedure and the time lost on nonsolvable inputs is often insignificant. Hence, it may be beneficial to execute this algorithm for all input groups that are not yet known to be nonsolvable.