Dear Elaheh Khamseh,
the Schur multiplier of a finite p-group (or of a polycyclic group in
general) can be computed in GAP with the function 'SchurMultiplicator'
of the Polycyclic package. This returns the abelian invariants of the
SchurMultiplier. It requires a PcpGroup as input.
The groups of order p^5 are available as part of the SmallGroups library of
GAP for every prime p. It is also not difficult to construct your desired
groups directly; see for example the function 'SpecialSplitExtensions'
enclosed below.
I hope that this helps you towards your questions and towards an
investigation of the Schur multipliers of your considered groups.
Best wishes,
Bettina
SpecialSplitExtensions := function(p)
local cl, grps;
# get conjugacy classes of elements of order p in GL(4,p)
cl := ConjugacyClasses(GL(4,p));
cl := List(cl, Representative);
cl := Filtered(cl, x -> Order(x) in [1,p]);
# construct a split extension for every conjugacy class
return List(cl,
x -> SplitExtensionPcpGroup(AbelianPcpGroup(1,[p]),[x]));
end;
gap> SpecialSplitExtensions(2);
[ Pcp-group with orders [ 2, 2, 2, 2, 2 ],
Pcp-group with orders [ 2, 2, 2, 2, 2 ],
Pcp-group with orders [ 2, 2, 2, 2, 2 ] ]
gap> List(last, SchurMultiplicator);
[ [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2 ] ]
gap> SpecialSplitExtensions(3);
[ Pcp-group with orders [ 3, 3, 3, 3, 3 ],
Pcp-group with orders [ 3, 3, 3, 3, 3 ],
Pcp-group with orders [ 3, 3, 3, 3, 3 ],
Pcp-group with orders [ 3, 3, 3, 3, 3 ] ]
gap> List(last, SchurMultiplicator);
[ [ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 ], [ 3, 3, 3, 3, 3, 3, 3 ],
[ 3, 3, 3, 3, 3, 3 ], [ 3, 3, 3 ] ]
gap> SpecialSplitExtensions(5);
[ Pcp-group with orders [ 5, 5, 5, 5, 5 ],
Pcp-group with orders [ 5, 5, 5, 5, 5 ],
Pcp-group with orders [ 5, 5, 5, 5, 5 ],
Pcp-group with orders [ 5, 5, 5, 5, 5 ],
Pcp-group with orders [ 5, 5, 5, 5, 5 ] ]
gap> List(last, SchurMultiplicator);
[ [ 5, 5, 5, 5, 5, 5, 5, 5, 5, 5 ], [ 5, 5, 5, 5, 5, 5, 5 ],
[ 5, 5, 5, 5, 5, 5 ], [ 5, 5, 5, 5 ], [ 5, 5, 5 ] ]
On Mon, 13 Sep 2010, Elaheh khamseh wrote:
Dears
Let G be a semidirect product of a normal subgroup elementery abelian
group of order p^4 and a cyclic group of order p. How can I copmute
its Schur multiplier ?
Do we have any information about the Schur multiplier of non-abelian
p-groups of order p^5?
Yours;
E. Khamseh.
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