Quotient Groups#

A powerful technique for investigating an fp-group is to study some of its quotients. For example, the action of a group \(G\) on the cosets of a subgroup \(H\) (where the index of \(H\) in \(G\) is finite) as described in the section on subgroups is an epimorphism of \(G\) onto a permutation group of degree \([G:H]\). In this section further methods for constructing various types of quotient group are described. Specifically, machinery is provided for constructing abelian quotients, \(p\)-quotients, nilpotent quotients, soluble quotients and \(PSL(2, q)/PGL(2, q)\) quotients.

Abelian Quotient#

Computing the quotient of an fp-group \(G\) by its derived group is usually the first step in analyzing an fp-group. The ability to do this to finite-index subgroups of \(G\) is a key step when trying to prove that a group is infinite. The intrinsics in this section compute information about the abelian quotient of an fp-group \(G\).

Intrinsics:

AbelianQuotient \((G)\) : The maximal abelian quotient \(G/G^\prime\) of the fp-group \(G\) is returned as a group in category GrpAb (cf. chapter Abelian Groups). The natural epimorphism \(\pi:G\rightarrow G/G^\prime\) is returned as second value.

HasFiniteAbelianQuotient \((G)\) : Return true if the maximal abelian quotient of the fp-group \(G\) is finite.

ElementaryAbelianQuotient \((G,~p)\) : The maximal \(p\)-elementary abelian quotient \(Q\) is returned as a group \(G\) in category GrpAb (cf. chapter Abelian Groups). The natural epimorphism \(\pi:G\rightarrow Q\) is returned as second value.

AbelianQuotientInvariants \((G)\) : Let \(G\) be an fp-group or a subgroup of finite index of an fp-group. The elementary divisors of the derived quotient group \(G/G'\) are returned as a sequence of integers.

AbelianQuotientInvariants \((G,~n)\) : Let \(G\) be an fp-group, \(n\) a small integer and \(N\) the normal subgroup of \(G\) generated by the derived group \(G'\) together with all \(n\)-th powers of elements of \(G\). The elementary divisors of the quotient group \(G/N\), are returned as a sequence of integers.

AbelianQuotientPrimes \((G)\) : Return a sequence of primes containing all prime divisors in the order of \(G/G'\), if this is finite. Note that there may be primes in the return sequence which do not divide this order. If [ 0 ] is returned then \(G/G'\) is infinite. If [] is returned, then \(G\) is perfect.

HasComputableAbelianQuotient \((G)\) : Given an fp-group \(G\), this intrinsic tests whether the abelian quotient of \(G\) can be computed. If so, it returns the value true, the abelian quotient \(A\) of \(G\) and the natural epimorphism \(\pi:G\rightarrow A\). If the abelian quotient of \(G\) cannot be computed, the value false is returned.

HasInfiniteComputableAbelianQuotient \((G)\) : Given an fp-group \(G\), this function tests whether the abelian quotient of \(G\) can be computed and is infinite. If so, it returns the value true, the abelian quotient \(A\) of \(G\) and the natural epimorphism \(\pi:G\rightarrow A\). If the abelian quotient of \(G\) cannot be computed or if it is finite, the value false is returned.

Notes:

(i): A number of variations on the above intrinsics may be found in chapter Finitely Presented Groups.
(ii): The above intrinsics require either a presentation for \(G\), or the coset table for \(G\) inside some supergroup of \(G\) which has a presentation. If such a coset table cannot be computed, an error will result.
(iii): Some functions may require the computation of a coset table. Experienced users can control the behaviour of an implicit coset enumeration by setting global parameters which can be changed using the function SetGlobalTCParameters.
(iv): Intrinsics with names of the form AbelianQuotientName can be abbreviated to AQName, where name is Quotient, Primes or Invariants.
(v): Considerable effort has been put into computing the Smith Normal Form for large integer matrices that can arise when executing these intrinsics. An important application where such large matrices arise is when the input is the integer matrix resulting from the abelianisation of the presentation of a subgroup \(H\) which has been constructed by rewriting the presentation of its parent group. If the index is large (even 50 or 100) then the presentation of \(H\) may be huge.
(vi): The intrinsic HasInfiniteComputableAbelianQuotient first checks the modular abelian invariants for a set of small primes. If for one of these primes the modular abelian quotient is trivial, then \(A\) must be finite and the function returns without actually computing the abelian quotient. This heuristic may save time if only infinite quotients are of interest.

Example: F27 (ex-12a8db)#

The Fibonacci group \(F(7)\) has the following \(2\)-generator presentation:

\[<a,b \ |\ a^2b^{-2}a^{-1}b^{-2}(a^{-1}b^{-1})^2, abab^2abab^{-1}(ab^2)^2>.\]

The goal is to determine the structure of this group.

> F<a, b> := FreeGroup(2);
> F7<a, b> := quo< F |  a^2*b^-2*a^-1*b^-2*(a^-1*b^-1)^2,
>        a*b*a*b^2*a*b*a*b^-1*(a*b^2)^2 >;
> F7;
Finitely presented group F7 on 2 generators
Relations
       a^2 * b^-2 * a^-1 * b^-2 * a^-1 * b^-1 * a^-1 * b^-1 =Id(F7)
       a * b * a * b^2 * a * b * a * b^-1 * a * b^2 * a * b^2 = Id()7

Quotient Groups#

Abelian Quotient#

\(p\)-Quotient#

Nilpotent Quotient#

Soluble Quotient#

Simple Group Quotients#

\({\rm L}_2\)-Quotients#