Semicharacters of Groups

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).

Key Words:

• Semicharacters

2010 Mathematics Subject Classification:

• 20D99

Notes

Note that the formula in [1] should read and not as stated.

Note that remark 2.5 cited above is only valid if restricted to l > 2 or q ≡ 1mod4. Otherwise, the claim that if l|q − 1 then an l-Sylow subgroup of GL(n, q) may be embedded in the subgroup of monomial matrices may not be valid. For example, in GL(2, 3) there are only 8 monomial matrices while the 2-Sylow subgroups are of order 16.

Communicated by S. Sehgal.