Central: BoolElt Default: false
NumberRandom: RngIntElt Default: 100
CompletionCheck: UserProgram Default:
Given an involution g in G, this function returns the
centraliser C of g in G using an algorithm of John Bray [Bra00].
Since it is Monte Carlo, it may return only a subgroup of the centraliser.
If Central is true, the projective centraliser of g will be
constructed: its elements commute with g modulo the centre of G.
The optional argument CompletionCheck is a function which can be used
to determine when we have constructed the centraliser. It takes the
following arguments: the parent group G; the proposed centraliser C;
the involution g. By default, the algorithm completes when 20 generators
have been found for the centraliser or when NumberRandom elements have
been considered.
Randomiser: GrpRandProc Default:
Central: BoolElt Default: false
NumberRandom: RngIntElt Default: 100
CompletionCheck: UserProgram Default:
Given an involution g in a matrix group G together with a SLP w
corresponding to g, this function returns the centraliser C of g
in G and SLPs for the generators of C. The algorithm used is due to
John Bray [Bra00]. Since it is Monte Carlo, it may return a proper
subgroup of the centraliser. If Central is true, the projective
centraliser of g is constructed: its elements commute with g modulo
the centre of G.
The parameter Randomiser specifies the random process that is to be
used to construct the centraliser. By default Randomiser is the
value returned by RandomProcessWithWords (G). The SLP for g
must lie in the word group of this process. The optional argument
CompletionCheck is a function which can be used to determine
when we have constructed the centraliser. It takes four arguments:
the parent group G; the proposed centraliser C; the involution g;
and the list of the SLPs for the generators of C. By default, the
algorithm completes when 20 generators have been found for the centraliser
or when NumberRandom elements have been considered.
Randomiser: GrpRandProc Default:
MaxTries: RngIntElt Default: 100
This Monte Carlo algorithm attempts to construct an element c of
the group G which conjugates the involution x to the involution y.
The corresponding SLPs for x and y are wx and wy respectively.
If such an element c is found, then three values are returned:
true, c and the SLP for c. Otherwise, the boolean value
false is returned. At most MaxTries random elements are
considered.
The parameter Randomiser specifies the random process to be used.
By default Randomiser is the value returned by
RandomProcessWithWords (G). The SLPs for x and y must lie
in the word group of this process and the SLP for c will also lie in
this word group.
NormalClosureMonteCarlo(G, H : parameters) : GrpPerm, GrpPerm -> GrpMat
slpsH: [] Default: []
ErrorProb: FltRatElt Default: 9/10
SubgroupChainLength: RngIntElt Default: Degree(H)
This Monte Carlo algorithm constructs the normal closure N of H in
G. If SLPs of the generators of H in the generators of G are
supplied via the parameter slpsH, then the function also returns
SLPs for the generators of N. The parameter SubgroupChainLength
is used to specify an upper bound on the length of any subgroup chain
in H. The probability that N is a proper subgroup of the normal
closure is bounded above by ErrorProb, assuming that
SubgroupChainLength is correctly set.
Randomiser: GrpRandProc Default:
NumberGenerators: RngIntElt Default: 10
MaxGenerators: RngIntElt Default: 100
Given a matrix group G defined over a finite field, this intrinsic
returns the derived group of G, and a list of SLPs of its generators
in the generators of G. The SLPs are elements of the word group of
the random process. The algorithm is Monte Carlo and may return a
proper subgroup of the derived group. The parameter Randomiser
specifies the random process to be used. By default Randomiser
is the value returned by RandomProcessWithWords (G). At least
NumberGenerators and at most MaxGenerators will be
constructed for the derived group.
NumberRandom: RngIntElt Default: 100
This intrinsic attempts to prove that a matrix or permutation group G
is perfect by establishing that its generators are in G'. Since it is
Monte-Carlo, there is a small probability of error. If the function
returns true, then G is perfect; if it returns false,
then G might still be perfect. Each call considers NumberRandom
random elements.
The algorithm is due to Leedham-Green and O'Brien [LGO02] and
it uses NormalSubgroupRandomElement.
We illustrate IsProbablyPerfect with a subgroup of GU(4, 9).
> G := GU(4, 9);
> N := sub<G | (G.1, G.2)>;
The generators of N have been chosen to be a normal generating set
for the derived group of G.
> IsProbablyPerfect(N);
false
> x := NormalSubgroupRandomElement(G, N);
> x;
[$.1^68 $.1^34 $.1^26 $.1^55]
[$.1^23 $.1^78 $.1^16 $.1^72]
[$.1^42 $.1^2 $.1^24 2]
[$.1^11 $.1^66 $.1^13 $.1^29]
> L := sub< G | N, x>;
> IsProbablyPerfect(L);
true
We now consider SO(7, 5) and Ω(7, 5).
> G := SO(7, 5);
> IsProbablyPerfect(G);
false
> G := Omega(7, 5);
> IsProbablyPerfect(G);
true
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