Torsion group
Torsion group with a central cyclic commutator subgroup
Let be a torsion group with exponent such that its commutator subgroup is central and cyclic
Properties
Representations
If the field contains an th root of unity, and
is a faithful central character of ,
then there exists a unique (up to equivalence) irrep with central character ,
and is itself faithful.1 #m/thm/group
If is a maximal abelian subgroup and is a linear character extending , then
where and denote corresponding -modules and denotes the induced module.
Moreover
[!check]- Proof
Let and , whence the Central extension of an abelian group
with associated commutator map .
Now is nondegenerate,
for if then
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