  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).
