[[Group character]]
# Central character

Given a group $G$, a **central character** $\chi : Z(G) \to \mathbb{K}^\times$ is a [[1-dimensional irrep|linear character]] for its [[Centre of a group|centre]]. #m/def/rep
A general [[group representation]] $\Gamma : G \to \opn{GL}(V)$ over $\mathbb{K}$ is said to **have central character** iff $Z(G)$ is represented as the [[centre of the general linear group]] $Z(V) \cong \mathbb{K}^\times$,
in which case the corresponding homomorphism is the corresponding central character.

## Properties

- [[Torsion group with a central cyclic commutator subgroup]]

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