58 Automorphism Groups of Special Ag Groups

This chapter describes functions which compute and display information about aut groups of finite soluble groups.

The algorithm used for computing the aut group requires that the soluble group be given in terms of a special ag presentation. Such presentations are described in the chapter of the GAP manual which deals with Special Ag Groups. Given a group presented by an arbitrary ag presentation, a special ag presentation can be computed using the function SpecialAgGroup.

The aut group is returned as a standard GAP group record. Auts are represented by their action on the sag group generating set of the input group. The order of the aut group is also computed.

The performance of the aut group algorithm is highly dependent on the structure of the input group. Given two groups with the same sequence of LG-series factor groups it will usually take much less time to compute the aut group of the one with the larger aut group. For example, it takes less than 1 second (Sparc 10/52) to compute the aut group of the exponent 7 extraspecial group of order 7^3. It takes more than 40 seconds to compute the aut group of the exponent 49 extraspecial group of order 7^3. The orders of the aut groups are 98784 and 2058 respectively. It takes only 20 minutes (Sparc 10/52) to compute the aut group of the 2-generator Burnside group of exponent 6, a group of order 2^{28}cdot 3^{25} whose aut group has order 2^{40}cdot 3^{53}cdot 5cdot 7; note, however, that it can take substantially longer than this to compute the aut groups of some of the groups of order 64 (for nilpotent groups one should use the function AutomorphismsPGroup from the ANU PQ package instead).

The following section describes the function that computes the aut group of a sag (see AutGroupSagGroup). It is followed by a description of Automorphism Group Elements and Operations for Automorphism Group Elements). Functions for obtaining some structural information about the aut group are described next (see AutGroupStructure, AutGroupFactors and AutGroupSeries). Finally, a function that converts the aut group into a form which may be more suitable for some applications is described (see AutGroupConverted).

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