- Outline of the algorithm
- The initialisation step
- Stabilisers in matrix groups
- Searching for a small generating set
- An interactive version of the algorithm
- Acknowledgements

A number of choices can be made by the user to influence the performance
of `AutomorphismGroupPGroup`

. Below we identify these choices
and their default values used in `AutomorphismGroup`

. We use the optional
argument `flag` of `AutomorphismGroupPGroup`

to invoke user-defined choices.
The possible values for `flag` are

`flag``= false`

- the user-defined defaults are employed in the algorithm.
See below for a list of possibilities.
`flag``= true`

- invokes the interactive version of the algorithm as described in Section An interactive version of the algorithm.

In the next section we give a brief outline of the algorithm which is necessary to understand the possible choices. Then we introduce the choices the later sections of this chapter.

The basic algorithm proceeds by induction
down the lower *p*-central series of a given *p*-group *G*; that is, it
successively computes *Aut*(*G*_{i}) for the quotients *G*_{i} = *G* / *P*_{i}(*G*) of
the descending sequence of subgroups defined by *P*_{1}(*G*) = *G* and
*P*_{i+1}(*G*)=[*P*_{i}(*G*),*G*] *P*_{i}(*G*)^{p} for *i* ≥ 1. Hence, in the initial
step of the algorithm, *Aut*(*G*_{2}) = *GL*(*d*,*p*) where *d* is the rank of
the elementary abelian group *G*_{2}. In the inductive step it determines
*Aut*(*G*_{i+1}) from *Aut*(*G*_{i}). For this purpose we introduce
an action of *Aut*(*G*_{i}) on a certain elementary abelian *p*-group *M*
(the ** p-multiplicator** of

Our package incorporates a number of refinements designed to simplify this stabiliser computation. Some of these refinements incur overheads and hence they are not always invoked. The features outlined below allow the user to select them.

Observe that the initial step of the algorithm returns *GL*(*d*,*p*). But
*Aut*(*G*) may induce on *G*_{2} a proper subgroup, say *K*, of *GL*(*d*,*p*).
Any intermediate subgroup of *GL*(*d*,*p*) which contains *K* suffices for
the algorithm and we supply two methods to construct a suitable subgroup:
these use characteristic subgroups or invariants of normal subgroups of *G*.
(See Section The initialisation step.)

In the inductive step an action of *Aut*(*G*_{i}) on an elementary abelian
group *M* is used. This action is computed as a matrix action on a vector
space. To simplify the orbit-stabiliser computation of the subspace *U*
of *M*, we can construct the stabiliser of *U* by iteration over a sequence
of *Aut*(*G*_{i})-invariant subspaces of *M*.
(See Section Stabilisers in matrix groups.)

Orbit-stabiliser computations in finite solvable groups given by a
polycyclic generating sequence are much more efficient than generic
computations of this type. Thus our algorithm makes use of a large
solvable normal subgroup *S* of *Aut*(*G*_{i}). Further, it is useful if
the generating set of *Aut*(*G*_{i}) outside *S* is as small as possible.
To achieve this we determine a permutation representation of *Aut*(*G*_{i})/*S*
and use this to reduce the number of generators if possible. (See Section
Searching for a small generating set.)

Assume we seek to compute the automorphism group of a *p*-group *G*
having Frattini rank *d*. We first determine as small as possible a
subgroup of *GL*(*d*, *p*) whose extension can act on *G*.

The user can choose the initialisation routine by assigning
`InitAutGroup`

to any one of the following.

`InitAutomorphismGroupOver`

- to use the minimal overgroups;
`InitAutomorphismGroupChar`

- to use the characteristic subgroups;
`InitAutomorphismGroupFull`

- to use the full
*GL*(*d*,*p*).

**a) Minimal Overgroups**

We determine the minimal over-groups of the Frattini subgroup of
*G* and compute invariants of these which must be respected by the
automorphism group of *G*. We partition the minimal overgroups and
compute the stabiliser in *GL*(*d*, *p*) of this partition.

The partition of the minimal overgroups is computed using the
function `PGFingerprint( G, U )`

. This is the time-consuming
part of this initialisation method. The user can
overwrite the function `PGFingerprint`

.

**b) Characteristic Subgroups**

Compute a generating set for the stabiliser in *GL* (*d*, *p*) of
a chain of characteristic subgroups of *G*. In practice, we construct
a characteristic chain by determining 2-step centralisers and omega
subgroups of factors of the lower *p*-central series.

However, there are often other characteristic subgroups which are not
found by these approaches. The user can overwrite the function
`PGCharSubgroups( G )`

to supply a set of characteristic subgroups.

**c) Defaults**

In the method for `AutomorphismGroup`

we use a default strategy:
if the value [(*p*^{d}−1)/(*p*−1)] is less than 1000, then we
use the minimal overgroup approach, otherwise the characteristic
subgroups are employed. An exception is made for homogeneous abelian
groups where we initialise the algorithm with the full group *GL*(*d*,*p*).

Consider the *i*th inductive step of the algorithm. Here *A* ≤ *Aut*(*G*_{i}) acts as matrix group on the elementary abelian *p*-group
*M* and we want to determine the stabiliser of a subgroup *U* ≤ *M*.

We use the MeatAxe to compute a series of *A*-invariant subspaces
through *M* such that each factor in the series is irreducible as
*A*-module. Then we use this series to break the computation
of *Stab*_{A}(*U*) into several smaller orbit-stabiliser calculations.

Note that a theoretic argument yields an *A*-invariant subspace
of *M* a priori: the nucleus *N*. This is always used to split
the computation up. However, it may happen that *N* = *M* and hence
results in no improvement.

`CHOP_MULT V`

The invariant series through *M* is computed and used if the
global variable `CHOP_MULT`

is set to `true`

. Otherwise, the algorithm
tries to determine *Stab*_{A}(*U*) in one step. By default, `CHOP_MULT`

is `true`

.

After each step of the computation, we attempt to find a nice generating set for the automorphism group of the current factor.

If the automorphism group is soluble, we store a polycyclic generating
set; otherwise, we store such a generating set for a large soluble
normal subgroup *S* of the automorphism group *A*, and as few generators
outside as possible. If *S* = *A* and a polycyclic generating set for
*S* is known, many steps of the algorithm proceed more rapidly.

`NICE_STAB V`

It may be both time-consuming and difficult to reduce the number of
generators for *A* outside *S*. Note that if the initialisation of the
algorithm is by `InitAutomorphismGroupOver`

, then we always know a
permutation representation for *A*/*S*. Occasionally the search for
a small generating set is expensive. If this is observed, one
could set the flag `NICE_STAB`

to `false`

and the algorithm no
longer invokes this search.

The choice of initialisation and the choice of chopping of the
*p*-multiplicator can also be driven by an interactive version
of the algorithm. We give an example.

gap> G := SmallGroup( 2^8, 1000 );; gap> SetInfoLevel( InfoAutGrp, 3 ); gap> AutomorphismGroupPGroup( G, true ); #I step 1: 2^3 -- init automorphisms choose initialisation (Over/Char/Full): # we choose Full #I init automorphism group : Full #I step 2: 2^3 -- aut grp has size 168 #I computing cover #I computing matrix action #I computing stabilizer of U #I dim U = 3 dim N = 6 dim M = 6 chop M/N and N: (y/n): # we choose n #I induce autos and add central autos #I step 3: 2^2 -- aut grp has size 12288 #I computing cover #I computing matrix action #I computing stabilizer of U #I dim U = 6 dim N = 5 dim M = 8 chop M/N and N: (y/n): # we choose y #I induce autos and add central autos #I final step: convert rec( glAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f3, f3, f4, f5, f6*f7, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f3*f5*f6, f2*f3, f3, f4, f5*f8, f6*f7, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f3, f2*f4, f3, f4*f7, f5*f7, f6*f7*f8, f7, f8 ] ], glOrder := 4, agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f4, f2, f3, f4*f8, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f4, f3, f4*f7, f5, f6*f7*f8, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f5, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f5, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f5, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f6, f2, f3, f4, f5*f7*f8, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f6, f3, f4*f7*f8, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f8, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f8, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f8, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f7, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f7, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f7, f4, f5, f6, f7, f8 ] ], agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], one := IdentityMapping( <pc group of size 256 with 8 generators> ), group := <pc group of size 256 with 8 generators>, size := 32768 )

Two points are worthy of comment.
First, the interactive version of the algorithm permits the user to
make a suitable choice in each step of the algorithm instead of making
one choice at the beginning. Secondly, the output of the `Info`

function
shows the ranks of the *p*-multiplicator and allowable subgroup,
and thus allow the user to observe the scale of difficulty
of the computation.

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.

[Up] [Previous] [Next] [Index]

AutPGrp manual

July 2018