Semicharacters of Groups

TL;DR

This paper introduces semicharacters of groups, explores their properties, and proves a conjecture about their order for specific finite groups like symmetric groups and GL(2,q).

Contribution

It defines semicharacters, formulates a conjecture about their order, and proves it for key classes of finite groups.

Findings

Conjecture holds for symmetric groups

Conjecture holds for GL(2,q) groups

Semicharacters generalize group homomorphisms

Abstract

We define the notion of a semicharacter of a group G : A function from the group to C*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is divisible by the order of the group. We prove that the conjecture holds for some important families of groups, including the Symmetric groups and the groups GL(2,q).