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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Stefan Kohl.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains constructors for the Suzuki groups.
##
## The generators are taken from
##
## Michio Suzuki. On a Class of Doubly Transitive Groups,
## Ann. Math. 75 (1962), 105-145.
##
## See the middle of page 140, in the proof of Theorem 12.
##
#############################################################################
##
#M SuzukiGroupCons( <IsMatrixGroup>, <q> )
##
InstallMethod( SuzukiGroupCons,
"matrix group for finite field size",
true,
[ IsMatrixGroup and IsFinite,
IsInt and IsPosRat ],
0,
function ( filter, q )
local G,f;
if not IsPrimePowerInt(q)
or SmallestRootInt(q) <> 2 or LogInt(q,2) mod 2 = 0
then Error("<q> must be a non-square power of 2"); fi;
f := GF(q);
G := GroupByGenerators(
[ImmutableMatrix(f,
[[1, 0, 0,0],
[1, 1, 0,0],
[1+Z(q), 1, 1,0],
[1+Z(q)+Z(q)^RootInt(2 * q),Z(q),1,1]] * One(f),true),
ImmutableMatrix(f,
[[0,0,0,1],
[0,0,1,0],
[0,1,0,0],
[1,0,0,0]] * One(f),true)]);
SetName(G,Concatenation("Sz(",String(q),")"));
SetDimensionOfMatrixGroup(G,4);
SetFieldOfMatrixGroup(G,f);
SetIsFinite(G,true);
SetSize(G,q^2*(q-1)*(q^2+1));
SetIsSimpleGroup(G, q > 2);
SetIsPerfectGroup(G, q > 2);
return G;
end );
#############################################################################
##
#M SuzukiGroupCons( <IsPermGroup>, <q> )
##
InstallMethod( SuzukiGroupCons,
"permutation group for finite field size",
true,
[ IsPermGroup and IsFinite,
IsInt and IsPosRat ],
0,
function ( filter, q )
local G,Ovoid,f,r,a,b,v;
if not IsPrimePowerInt(q)
or SmallestRootInt(q) <> 2 or LogInt(q,2) mod 2 = 0
then Error("<q> must be a non-square power of 2"); fi;
f := GF(q);
r := RootInt(2 * q);
v := [1,0,0,0] * One(f);
v := ImmutableVector(f, v);
Ovoid := [v];
for a in f do
for b in f do
v:=[a^(r+2) + a*b + b^r,b,a,One(f)];
v := ImmutableVector(f, v);
Add(Ovoid,NormedRowVector(v));
od;
od;
Sort(Ovoid);
G := Action(SuzukiGroupCons(IsMatrixGroup,q),Ovoid,OnLines);
SetName(G,Concatenation("Sz(",String(q),")"));
SetSize(G,q^2*(q-1)*(q^2+1));
SetIsSimpleGroup(G, q > 2);
SetIsPerfectGroup(G, q > 2);
return G;
end );
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