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
= false
= true 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(Gi) for the quotients Gi = G / Pi(G) of the descending sequence of subgroups defined by P1(G) = G and Pi+1(G)=[Pi(G),G] Pi(G)p for i ≥ 1. Hence, in the initial step of the algorithm, Aut(G2) = GL(d,p) where d is the rank of the elementary abelian group G2. In the inductive step it determines Aut(Gi+1) from Aut(Gi). For this purpose we introduce an action of Aut(Gi) on a certain elementary abelian p-group M (the p-multiplicator of Gi). The main computation of the inductive step is the determination of the stabiliser in Aut(Gi) of a subgroup U of M (the allowable subgroup for Gi+1). This stabiliser calculation is the bottle-neck of the algorithm.
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 G2 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(Gi) 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(Gi)-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(Gi). Further, it is useful if the generating set of Aut(Gi) outside S is as small as possible. To achieve this we determine a permutation representation of Aut(Gi)/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
InitAutomorphismGroupChar
InitAutomorphismGroupFull 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 [(pd−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 ith inductive step of the algorithm. Here A ≤ Aut(Gi) 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 StabA(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 StabA(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.
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AutPGrp manual