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Most of the functions described here use data derived from the Web Atlas. The
data has been prepared for inclusion in Magma by Michael Downward and
Eamonn O'Brien. It maintains Atlas names, conventions and orderings.
All of these functions, except GoodBasePoints, accept as input matrix or
permutation groups. The algorithm underpinning GoodBasePoints due to
O'Brien & Wilson [OW05].
Projective: BoolElt Default: false
AutomorphismGroup: BoolElt Default: false
Construct standard generators for small quasisimple or sporadic group G
having name str;
words in SLP group defined on the defining generators of G are
also obtained for the standard generators.
If G is sporadic and AutomorphismGroup is
true, assume G is automorphism group of group having name
str.
If standard generators found, return true and sequences of
generators and corresponding words, else false.
Note: A return value of
false only means that the algorithm's random search for standard
generators did not succeed within the number of tries allowed. If the user
is sure the group G matches the name str,
then they should try the function again.
If G is absolutely irreducible matrix group
and Projective is true, then construct
standard generators possibly modulo centre of G.
This function currently works for all sporadic simple groups and all
quasisimple groups for which the simple quotient has order at most
2 x 108. If you call it with an invalid value of str, then it will
print out a list of all valid values.
A list of valid strings for the second argument of StandardGenerators.
The standard copy of the group G having the name str. If the second
return value is true, then the group H returned is a matrix group
with nontrivial scalar subgroup Z, and it is H/Z rather than H that
is isomorphic to G.
Projective: BoolElt Default: false
AutomorphismGroup: BoolElt Default: false
Use the StandardGenerators function to construct a (possibly projective)
isomorphism from G to a standard copy of G. Options as for
StandardGenerators. The first returned value indicates whether the call of
StandardGenerators was successful.
Projective: BoolElt Default: false
Generators: SeqEnum Default: []
AutomorphismGroup: BoolElt Default: false
Return true if standard presentation is satisfied by generators of
sporadic group G having name str, else false.
If AutomorphismGroup is
true, assume G is automorphism group of sporadic group having
name str.
Standard generators may be supplied as Generators, otherwise
defining generators are assumed to be standard.
If G is absolutely irreducible matrix
group and Projective is true, then
verify presentation modulo centre of G.
Projective: BoolElt Default: false
Generators: SeqEnum Default: []
AutomorphismGroup: BoolElt Default: false
Construct some maximal subgroups for sporadic group G having
name str. If AutomorphismGroup is
true, assume G is automorphism group of sporadic group having
name str and construct some of its maximal subgroups.
If standard generators supplied as Generators or found for G then
return true and list of subgroups, else return false.
If G is absolutely irreducible matrix group and Projective
is true, then construct standard generators and so
subgroups possibly modulo centre of G.
Projective: BoolElt Default: false
Generators: SeqEnum Default: []
Construct certain subgroups for sporadic group G having
name str. If standard generators supplied as
Generators or found for G then
return true and list of subgroups, else return false.
If G is absolutely irreducible matrix
group and Projective is true, then construct standard generators
possibly modulo centre of G.
Projective: BoolElt Default: false
Generators: SeqEnum Default: []
If standard generators supplied as Generators
or found for sporadic group G having
name str, then return true and list of
base points for G, else return false.
If G is absolutely irreducible and Projective is true,
then standard generators are possibly modulo centre of G, and base
points are correspondingly adjusted.
Display stored subgroup data for sporadic group having name str.
AutomorphismGroup: BoolElt Default: false
Display stored data for some maximal subgroups of
sporadic group having name str.
If AutomorphismGroup is
true, then display stored data for some maximal subgroups of
automorphism group of sporadic group.
The machinery is illustrated in the case of the sporadic Janko group J1.
> G :=
> MatrixGroup<7, GF(11) |
> [ 9, 1, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 9, 1, 3, 1, 3, 3, 9, 1, 3, 1, 3,
> 3, 9, 1, 1, 1, 3, 3, 9, 1, 1, 3, 3, 3, 9, 1, 1, 3, 1, 3, 9, 1, 1, 3, 1, 3 ],
> [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0,
> 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 10, 10, 0, 0, 0, 0, 0,0] >;
> flag, S := StandardGenerators (G, "J1");
> flag;
true
> StandardPresentation (G, "J1": Generators := S);
true
> flag, M:= MaximalSubgroups (G, "J1": Generators := S);
> #M;
7
> M[4];
rec<recformat<name: MonStgElt, parent: MonStgElt, generators: SeqEnum,
group: Grp, order: RngIntElt, index: RngIntElt> |
name := 19:6,
parent := J1,
group := MatrixGroup(7, GF(11))
Generators:
[ 0 1 4 3 3 4 7]
[ 1 2 8 3 6 2 9]
[ 4 8 10 1 6 0 9]
[ 3 3 1 8 9 1 10]
[ 3 6 6 9 1 3 7]
[ 4 2 0 1 3 0 9]
[ 7 9 9 10 7 9 0]
[ 4 6 2 3 8 1 6]
[ 8 1 3 10 2 7 4]
[ 3 6 1 0 6 9 6]
[ 2 3 6 9 0 3 7]
[ 7 8 5 2 4 6 4]
[10 4 5 2 8 6 8]
[10 9 0 1 9 8 9],
order := 114,
index := 1540
>
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