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4 Influencing the algorithm

Sections

  1. Outline of the algorithm
  2. The initialisation step
  3. Stabilisers in matrix groups
  4. Searching for a small generating set
  5. An interactive version of the algorithm
  6. 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.

4.1 Outline of the algorithm

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.)

4.2 The initialisation step

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 [(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).

4.3 Stabilisers in matrix groups

Consider the ith inductive step of the algorithm. Here AAut(Gi) acts as matrix group on the elementary abelian p-group M and we want to determine the stabiliser of a subgroup UM.

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.

    4.4 Searching for a small generating set

    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.

    4.5 An interactive version of the algorithm

    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.

    4.6 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.

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    AutPGrp manual
    July 2018