[GAP Forum] Characters

[Thomas Breuer]

Dear GAP Forum,

D. Naidu wrote

> I hope you will excuse me for posting the following non-GAP-related question:
> 
> Let G be a finite group and let V be a complex finite dimensional representation of G with character \chi.
> 
> Let Ker(\chi)  = {g \in G   |    \chi(g) = deg(\chi)} and let
> |Ker(\chi)| = {g \in G  |  |\chi(g)| = deg(\chi)}.
> 
> It is well know that the above two sets are equal to the sets
> {g \in G  |  g acts as identity on V},
> {g \in G  |  g acts by a scalar on V}, respectively.
> 
> Now let H be a normal subgroup of G.
> 
> Then the set of irreducible character \chi of G such that H < Ker(\chi)
> is in one-to-one correspondence with irreducible characters of the
> quotient group G/H.
> 
> My question is: what can be said about the set of irreducible characters
> \chi  of G such that H < |Ker(\chi)|?
> What is H is abelian?

The set $\{ g \in G | |\chi(g)| = \chi(1) \}$ is usually called the
centre of $\chi$, and denoted by $Z(\chi)$.

For a normal subgroup $H$ of $G$,
we have the equality
\[
   \{ \chi \in Irr(G) | H \subseteq Z(\chi) \}
    = \{ \chi \in Irr(G) | [G,H] \subseteq \ker(\chi) \} ,
\]
where $[A,B]$ denotes the commutator subgroup of the groups $A$, and $B$,
i.e., the group generated by the commutators $a^{-1} b^{-1} a b$
with $a \in A, b \in B$.

So the set of characters in question is in bijection with the irreducible
characters of $G/[G,H]$,
which is the largest central extension of $G/H$ that occurs in $G$,
i.e., the largest factor group $F$ of $G$ such that
the kernel of the natural epimorphism from $F$ onto $G/H$ is central in $F$.

All the best,
Thomas