Database of Irreducible Soluble Subgroups of GL(n,p) for n > 1 and p^n < 256

Version 1.0 Date of release: October 1990

M. W. Short Mathematics Programme Murdoch University Murdoch, WA 6150 Australia E-mail: short@jordan.csu.murdoch.edu.au

Finite soluble groups can be uniformly described using polycyclic presentations. Such presentations were introduced by Jurgensen (1970), who used the term AG-system to describe them. We give only a brief description of them here.

A polycyclic generating sequence for a finite soluble group G is a sequence a_1, a_2, ..., a_n of elements of G such that the set of these elements generates G and <a_(i+1), ..., a_n> is normal in <a_i, ..., a_n>. Such a sequence gives rise to a presentation of G that exhibits the action of a_i on <a_(i+1), ..., a_n>; we call this a polycyclic presentation. Each element of G can then be uniquely represented as a sequence of n non-negative integers; we call such a sequence the normal form of the element. If H = <b_1, ..., b_r> is a subgroup of G, then we can represent H by storing the normal form of each b_i.

A maximal soluble subgroup of a group is a soluble subgroup that is not properly contained in any other soluble subgroup.

For certain pairs (n,p), Jordan (see Dieudonne (1961)) and Suprunenko (1976) developed methods which allow one to construct a set, S, of certain irreducible soluble subgroups of GL(n,p). This set has the property that every irreducible maximal soluble subgroup of GL(n,p) is conjugate to exactly one of its members. We define a JS-maximal (for Jordan-Suprunenko) to be any group belonging to S. Note that every JS-maximal is irreducible and soluble, but not necessarily maximal soluble. Clearly, every irreducible soluble subgroup of GL(n,p) is conjugate to a subgroup of at least one JS-maximal.

The database isolgps (formerly called irredsol) provides access to one representative of each conjugacy class of irreducible soluble subgroups of GL(n,p), p prime, for n > 1 and p^n < 256.

The other main feature is a procedure for constructing the semidirect product of a finite vector space and an irreducible matrix group over that space:

Getvecs(G)

This function takes a matrix group G over a finite prime field and returns a sequence, VECS say, containing all the vectors of the natural module for G. The ordering of VECS does NOT depend on G, but only on its natural module.

Semidir(G, VECS)

Given an irreducible matrix group G of degree n and over a finite prime field of character p and the sequence VECS obtained from Getvecs, this function returns the permutation group H of degree p^n that is the semidirect product of G with its natural module. H acts on the set {1 .. p^n} and G is isomorphic to each of the point stabilizers. It is well known that H is primitive, and that every primitive permutation group with soluble socle arises in this way. Note that if Semidir is to be called more than once for subgroups of the same GL(n, p), then Getvecs need only be called on the first occasion, since the ordering of VECS depends only on n and p. This is why the call to Getvecs is not made by Semidir itself.

Each group is labelled according to the dimension of the matrices and the size of the coefficient field. For example, the label of the 4th group of 3-dimensional matrices over GF(5) is 3, 5, 4.

The groups are stored internally in a compact representation, so a little setting up must be done to generate a complete matrix group. Some information about each group is also stored, so there are several ways of searching for groups with particular properties.

The groups are accessed through the following functions only:

IsolGroup(n, p, i)

This function returns the i-th group of degree n over GF(p).

IsolNumberOfDegreeField(n, p)

This function returns the number of groups of degree n over GF(p) stored in the library.

IsolInfo(n, p, i)

This function returns a string which gives some information about a group given its label. In particular, it contains the order and primitivity information.

IsolDegree(n, p, i)

This function returns the degree of a group given its label (i.e., n).

IsolCharacteristic(n, p, i)

This function returns the characteristic of the coefficient ring given its label (i.e., p).

IsolOrder(n, p, i)

This function returns the order of a group given its label.

IsolMinBlockSize(n, p, i)

This function returns the minimal block size of a group given its label. If it is primitive, it returns 0.

IsolIsPrimitive(n, p, i)

This function returns whether a group is primitive given its label.

IsolGuardian(n, p, i)

This function returns the "guardian" of a group given its label (i.e., the maximal subgroup of GL(n, p) of which the group is a subgroup). A predicate for a group in this library is one of the following:

• A function f: (<GrpMat>) --> BoolElt (which may either be an intrinsic function or a user defined function).
• A tuple of one function <g>, where g: (<RngIntElt>, <RngIntElt>, <RngIntElt>) --> BoolElt (again, either intrinsic or user defined). This returns a boolean given the label of a group.
• A tuple of two functions <g, f> where g, f as above. In this case, the tested predicate will be g(n, p, i) and f(IsolGroup(n, p, i)). This form is introduced to avoid expanding the group from its label until absolutely necessary.

IsolGroupSatisfying(f)

Given a predicate f, return a group satisfying f(G). This function runs through all the stored groups and applies the predicate f until it finds a suitable one. If no group is found, an error message is printed.

IsolGroupOfDegreeSatisfying(d, f)

As IsolGroupSatisfying(f), except it only runs through the groups of degree d.

IsolGroupOfDegreeFieldSatisfying(d, p, f)

As IsolGroupSatisfying(f), except it only runs through the groups of degree d and defined over GF(p).

IsolGroupsSatisfying(f)

As IsolGroupSatisfying(f), except a sequence of all such groups is returned.

IsolGroupsOfDegreeSatisfying(d, f)

As IsolGroupOfDegreeSatisfying(d, f), except a sequence of all such groups is returned.

IsolGroupsOfDegreeFieldSatisfying(d, p, f)

As IsolGroupOfDegreeFieldSatisfying(d, p, f), except a sequence of all such groups is returned.

IsolProcess()

Return a "process" for looping over all the stored groups. Initially it points to the first group (of degree 2 and over GF(2)). A specifier for degree or field is one of:

• A valid degree (field size)
• A tuple <lo, hi> of valid degrees (field sizes) which is interpreted to mean all degrees (prime field sizes) in [lo, hi].

IsolProcessOfDegree(d)

Return a "process" for looping over all the stored groups with degree specifier d. Initially it points to the first such group.

IsolProcessOfDegreeField(d, p)

Return a "process" for looping over all the stored groups with degree specifier d and field specifier p. Initially it points to the first such group (the principal key is the degree).

IsolProcessOfField(p)

Return a "process" for looping over all the stored groups with degree specifier p. Initially it points to the first such group (sorted again by degree first).

IsolProcessIsEmpty(P)

Return whether the process P currently points to a group.

IsolProcessGroup(P)

Given a process P which currently points to a group, return that group.

IsolProcessInfo(P)

Given a process P which currently points to a group, return the associated information string.

IsolProcessLabel(P)

Given a process P which currently points to a group, return the label n, p, i of the group.

IsolProcessNext(~P)

Given a process P which currently points to a group, modify it so that it points to the next group if there is one or make it empty if there is not.

Example

load isolgps;
P := IsolProcessOfDegree(2);
while not IsolProcessIsEmpty(P) do
    G := IsolProcessGroup(P);
    ... do computations with G ...
    IsolProcessNext(~P);
end while;

Subscreens

• Bibliography for Database of Irreducible Soluble Subgroups of GL(n,p) for n > 1 and p^n < 256
• Diagram of Contents of Database of Irreducible Soluble Subgroups of GL(n,p) for n > 1 and p^n < 256