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# AutPGrp

## Computing the Automorphism Group of a p-Group

1.12.0

17 May 2026

Bettina Eick
Email: beick@tu-bs.de
Homepage: http://www.iaa.tu-bs.de/beick
Address:
Institut Analysis und Algebra
TU Braunschweig
Universitätsplatz 2
D-38106 Braunschweig
Germany

Eamonn O'Brien
Email: obrien@math.auckland.ac.nz
Homepage: http://www.math.auckland.ac.nz/~obrien
Address:
Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
New Zealand

### Abstract

The AutPGrp package introduces 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 ANUPQ 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. 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.

### Copyright

Bettina Eick and Eamonn O'Brien.

AutPGrp is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

### Acknowledgements

We thank Alexander Hulpke for helping us with efficiency problems. Werner Nickel provided some functions from the GAP `PQuotient` which are used in this package.

### Contents

1 Introduction
1.1 Introduction
2 The automorphism group method
2.1 The automorphism group method
2.1-1 AutomorphismGroup
2.1-2 InfoAutGrp
3 The underlying function
3.1 The underlying function
3.1-1 AutomorphismGroupPGroup
3.1-2 ConvertHybridAutGroup
3.1-3 PcGroupAutPGroup
4 Influencing the algorithm
4.1 Outline of the algorithm
4.2 The initialisation step
4.3 Stabilisers in matrix groups
4.3-1 CHOP_MULT
4.4 Searching for a small generating set
4.4-1 NICE_STAB
4.5 An interactive version of the algorithm
5 Additional features of the package
5.1 Additional features of the package
5.1-1 NumberOfPClass2PGroups
5.1-2 NumberOfPClass2PGroups
5.1-3 NumberOfClass2LieAlgebras
5.1-4 NumberOfClass2LieAlgbras
Index
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