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 group If is a maximal abelian subgroup and is a linear character extending , then

where and denote corresponding [[Module over a group|-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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Footnotes

1. 1988. Vertex operator algebras and the Monster, §5.5, p. 118 ↩