GAP (AutPGrp) - Chapter 4: Influencing the algorithm

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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

### 4 Influencing the algorithm

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` (2.1-1). 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 4.5.

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 \(Aut(G_i)\) 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 \(i\geq 1\). 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 \(G_i\)). The main computation of the inductive step is the determination of the stabiliser in \(Aut(G_i)\) of a subgroup \(U\) of \(M\) (the allowable subgroup for \(G_{i+1}\)). 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 \(G_2\) 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 4.2.)

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 4.3.)

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 4.4.)

#### 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: 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`.

`InitAutomorphismGroupChar`

  to use the 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.

`InitAutomorphismGroupFull`

  to use the full \(GL(d,p)\).

In the method for `AutomorphismGroup` (2.1-1) we use a default strategy: if the value \(\frac{p^d-1}{p-1}\) 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 \(GL(d,p)\).

#### 4.3 Stabilisers in matrix groups

Consider the \(i\)th inductive step of the algorithm. Here \(A \leq Aut(G_i)\) acts as matrix group on the elementary abelian \(p\)-group \(M\) and we want to determine the stabiliser of a subgroup \(U \leq M\).

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.

##### 4.3-1 CHOP_MULT

`‣ CHOP_MULT` | ( global variable )

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`.

#### 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.

##### 4.4-1 NICE_STAB

`‣ NICE_STAB` | ( global variable )

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 := PcGroupCode(2504972218445939562621471375, 256);;  # 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.

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