Torsion group with a central cyclic commutator subgroup

Let $𝐺$ be a torsion group with exponent $𝑠$ such that its commutator subgroup is central and cyclic

[𝐺, 𝐺] \leq 𝑍 (𝐺) ≅ ℤ + 𝑠 0

Properties

Representations

If the field $𝕂 \times$ contains an $𝑠$ th root of unity, and

𝜒 : 𝑍 (𝐺) ↪ 𝕂 \times

is a faithful central character of $𝐺$ , then there exists a unique (up to equivalence) irrep $Γ : 𝐺 \to G L (𝑉)$ with central character $𝜒$ , and $Γ$ is itself faithful.¹ #m/thm/group If $𝐴 \leq 𝐺$ is a maximal abelian subgroup and $𝜓 : 𝐴 \to 𝕂 \times$ is a linear character extending $𝜒$ , then

𝑉 Γ = I n d 𝐺 𝐴 𝕂 𝜓 = 𝐺 \otimes 𝐴 𝕂 𝜓

where $𝑉 Γ$ and $𝕂 𝜓$ denote corresponding $𝐺$ -modules and $I n d 𝐺 𝐴 𝕂 𝜓$ denotes the induced module. Moreover

d i m 𝑉 = | 𝐴 / 𝑍 (𝐺) | = | 𝐺 / 𝐴 | = \sqrt | 𝐺 / 𝑍 (𝐺) |

[!check]- Proof Let $𝑍 (𝐺) = ⟨ 𝜅 ⟩$ and $𝑉 = 𝐺 / 𝑍 (𝐺)$ , whence the Central extension of an abelian group

1 \to ℤ + 𝑠 0 𝜅 ↪ 𝐺 𝜋 ↠ 𝑉 \to 1

with associated commutator map $𝑐 0 : 𝑉 \times 𝑉 \to ℤ 𝑠 0$ . Now $𝑐 0$ is nondegenerate, for if $[𝑣, 𝑉] = 1$ then

#state/develop | #lang/en | #SemBr

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

Torsion group with a central cyclic commutator subgroup

Properties

Representations

Footnotes