Conjugacy Classes in Solvable Groups

We defer the general discussion of conjugacy class computations to Section 9.4.

Here we only describe an algorithm from [Mecky and Neub¨user, 1989] to solve the following problem:

LetQG≤Sym(),with|| =n, and suppose thatQis elementary abelian. Given representatives of the conjugacy classes ofG/Qand

the centralizers of the representatives inG/Q, construct representatives for the conjugacy classes ofGand

the centralizers of the representatives.

(7.6) Let {Qh1, . . . ,Qhk}be the set of representatives for the conjugacy classes of G/Q, and let ¯Ci := CG/Q(Qhi) fori ∈ [1,k]. We denote the complete preimage of ¯CiinGbyCi.

Computation of the Class Representatives

For anyg∈G, the conjugacy class ofgintersects nontrivially exactly one of the cosets Qhi (namely, the representative of the conjugacy class of Qg in G/Q). Hence we can find representatives for the conjugacy classes ofG in the cosetsQh₁, . . . ,Qhk. For fixedi ∈[1,k], two elementsg₁,g₂∈ Qhi are conjugate inGif and only ifg2∈ g₁^Ci, since any element ofG\Ci conjugates g1outside of the cosetQhi. Therefore, we can obtain representatives for the conjugacy classes ofGthat intersectQhinontrivially by computing the orbits of the conjugation action ofCi on Qhiand taking a representative from each orbit.

The argument in the previous paragraph is valid for any normal subgroupQ ofG. However, we can exploit the fact thatQis elementary abelian and so can be considered as a vector space, allowing us to make two possible shortcuts in the foregoing computation. To simplify notation, we drop the indexi and consider an arbitrary cosetQh ∈ {Qh₁, . . . ,Qhk}, with ¯C :=CG/Q(Qh). We denote the complete preimage of ¯CinGbyC.

Forqh∈ Qhandc∈C, we ha ve (qh)^c=q^c[c,h⁻¹]h. For fixedc(andh), the mapAc:Q→ Qdefined by the ruleAc:q →q^c[c,h⁻¹] is a n a ffine ma p of the vector spaceQ, sinceq →q^cis a linear transformation and [c,h⁻¹] is a fixed element ofQ, independent ofq. Therefore the set of permutations ofQh, defined by the conjugation actions of the elements ofC, correspond to aset of permutations defined by the affine transformations{Ac|c∈C}ofQ. We claim that AC:= {Ac|c ∈ C}is in fact a group, and with the natural identification ϕ:Qh → Qdefined asϕ:qh → q, the conjugation action ofC on Qh is permutation isomorphic to the action ofAC onQ. Indeed,

q^c1c2[c₁c₂,h⁻¹]=(q^c1[c₁,h⁻¹])^c2[c₂,h⁻¹],

since [c1c2,h⁻¹]=[c1,h⁻¹]^c2[c2,h⁻¹]. This last identity is a special case of Lemma 2.3.10, and it can also be checked directly.

The first shortcut is to compute an SGS{z1, . . . ,zm}of Q supporting co-ordinatization and matrices in this basis for the affine transformations AT := {Ac|c ∈ T}for agenerating setT of C. Then, instead of the orbits of the conjugation ofC onQh, we compute the orbits of AC = ATonQand take the preimages of representatives of these orbits underϕ⁻¹. The affine images of elements of the vector space Qcan be computed much more quickly than the conjugates of the corresponding elements ofQh.

The second shortcut goes even further and computes the orbits of AC= ATon a potentially smaller factor space of Qinstead of Q. Since QC, Lemma6.1.7 implies that the orbits of the subgroupAQ:= {Ac|c∈ Q}ofAC

are permuted by AC. Forc ∈ Q, we ha veq^c = q for allq ∈ Q and so Ac

is just a translation of the vector space Q by the vector [c,h⁻¹]. Hence, for the 0 vector of Q, we ha ve 0^AQ = [Q,h⁻¹]. This subspace [Q,h⁻¹] ca n be computed easily, and the orbits ofAC onQare the complete preimages of the orbits of AC on the factor space Q/[Q,h⁻¹]. Let ¯AC denote the permutation action of AC onQ/[Q,h⁻¹]. Note that ¯AC acts on Q/[Q,h⁻¹] as a group of affine transformations as well, since for anyq,r∈ Qandc∈Cwe have

[q,h⁻¹]^c=(q⁻¹)^ch^cq^c(h⁻¹)^c=(q⁻¹)^ch[h,c]q^c(h⁻¹)^c

=(q⁻¹)^chq^c[h,c](h⁻¹)^c=(q⁻¹)^chq^ch⁻¹=[q^c,h⁻¹] and so ([Q,h⁻¹]r)^c[c,h⁻¹]=[Q,h⁻¹]r^c[c,h⁻¹].

Computation of the Centralizers

To apply (7.6) recursively, we also have to compute the centralizers of the repre-sentatives of the conjugacy classes inG. Let [Q,h⁻¹]q1, . . . ,[Q,h⁻¹]qlbe the representatives of the orbits of ¯AConQ/[Q,h⁻¹] (and soq₁h, . . . ,qlhare the representatives of those conjugacy classes ofGthat intersectQhnontrivially).

During the orbit computations inQ/[Q,h⁻¹], we can build Schreier treesSj

that code elements of ¯ACcarrying the representatives [Q,h⁻¹]qj,j∈[1,l], to other elements of their orbits (cf. the end of Section 2.1.1). The elements coded bySj comprise a transversal for ¯AC mod ( ¯AC)_[Q,h⁻¹]qj, and so some form of point stabilizer subgroup constructions can be used to obtain generators for D¯j :=( ¯AC)_[Q,h⁻¹]qj. Note that we know|D¯j|in advance, so the randomized methods from Chapter 4 give guaranteed correct output as well.

LetDjdenote the complete preimage of ¯DjinAC. We ha veAQ≤Dj, since AQacts trivially onQ/[Q,h⁻¹]. In terms of the action ofAC onQ, the group Djis the setwise stabilizer of [Q,h⁻¹]qj. We still have to constructCG(qjh), which is permutation isomorphic to the point stabilizer subgroup ofDjfixing qj. For that, we do not need another orbit computation on [Q,h⁻¹]qj; instead, we can use linear algebra.

As afirst step, we construct (AQ)q_j. For q∈Q, we ha ve Aq(qj)=q^q_j[q, h⁻¹], which is equal to qj if and only if [q,h⁻¹]=1, that is, q∈CQ(h)= CQ(qjh). Hence (AQ)qj can be obtained as the fixed point space ofAh, which is a linear transformation ofQ.

Next, we claim that, for anyc∈ Dj,

(i) there is an elementr(c)∈ Qsuch thatA_cr(c)fixesqj; a nd (ii) ifAcr₁,Acr₂fixqjfor somer1,r2 ∈Qthenr1r₂⁻¹∈CQ(h).

To show (i), observe that

Ac(qj)=q^c_j[c,h⁻¹]=[q,h⁻¹]qj (7.7) for someq ∈ Q, sinceAcfixes [Q,h⁻¹]qj. For thatq, using (7.7) and using repeatedly thatQis abelian and that [c,h⁻¹]∈ Q, we ha ve

Acq⁻¹(qj)=q^cq_j ⁻¹[cq⁻¹,h⁻¹]=q^c_j[c,h⁻¹]^q−1[q⁻¹,h⁻¹]

=q^c_j[c,h⁻¹][q⁻¹,h⁻¹]=[q,h⁻¹]qj[q⁻¹,h⁻¹]=qj. Hencer(c) :=q⁻¹ satisfies (i). Moreover, if Acr₁(qj) = Acr₂(qj) = qj then [cr₁,h⁻¹]=[cr₂,h⁻¹], implying [r₁,h⁻¹]=[r₂,h⁻¹] a ndh^r1r−12 =h. There-fore, (ii) holds.

We can constructCG(qjh) as follows: For each generatorcofDj = T, we constructr(c) satisfying (i), by solving the system of linear equations

q^c_j[c,h⁻¹]q⁻_j¹=q⁻¹·q^h−1

for the coordinates ofq in the basis{z1, . . . ,zm}of Qsupporting coordinati-zation. This equation was obtained by rewriting (7.7). The left-hand side is a known vector inQ, and the right-hand side can be expressed as a linear function of the coordinates ofq. As we have seen in the previous paragraph,r(c) :=q⁻¹ satisfies (i). LetCQ(h) = S. By (ii), the set S∪ {cr(c)|c ∈ T}generates CG(qjh).

The running time and memory requirement of the algorithm is proportional to|Q/[Q,h⁻¹]|, which may be exponentially large in terms of the input length.

In general, an exponential blowup is unavoidable, since the length of the output may be this large. This happens, for example, ifQ≤Z(G). Hence, the method should be used judiciously. For example, when computing the conjugacy classes of a solvable group, we may try to arrange the series of normal subgroups with elementary abelian factors guiding the recursion so that central extensions come at the end of the recursion. Moreover, we may have to restrict ourselves only to the counting of the conjugacy classes instead of listing representatives.