[GAP Forum] Characters

[D N]

Dear GAP Forum,

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?

Best regards,
D. Naidu

 
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