GAP (AutPGrp) - Chapter 1: Introduction

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1 Introduction
1.1 Introduction

### 1 Introduction

#### 1.1 Introduction

Given an arbitrary finite group, the computation of its automorphism group is a very difficult task. Pioneer work in this area was carried out by Felsch and Neubueser (1970), whose algorithm used the output of their subgroup lattice program. A technique developed by Neubueser in the early 1970s sought to compute the automorphism group viewed as a permutation group acting on unions of certain conjugacy classes of the group. A similar method was implemented by Hulpke (1997) in the GAP 4 library. Recently, Cannon and Holt (1999) presented a new algorithm which uses a hybrid group approach.

More efficient approaches are available to determine the automorphism group for groups satisfying certain properties. Following the work of Shoda (1928), Hulpke in 1997 implemented a practical method for finite abelian groups in the GAP 4 library. Wursthorn (1993) adapted modular group algebra techniques to compute the automorphism groups of p-groups; the GAP 3 share package Sisyphos includes an implementation. Smith (1994) introduced an algorithm for finite solvable groups which is available in the AutAg share package of GAP 3.

Moreover, the p-group generation method of Newman (1977) and O'Brien (1990) can be modified to compute the automorphism group of a finite p-group as outlined in O'Brien (1995). This algorithm is implemented in the ANU `pq` C program.

Here we introduce a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANU `pq` method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions.

The GAP 4 package ANUPQ, which is an interface to most of the functionality of the ANU `pq` C program, uses the AutPGrp package to compute automorphism groups of p-groups.

We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed.

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