# Character Theory

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• May 11th 2011, 01:46 PM
Turloughmack
Character Theory
Urgent answer required for these questions please!

(1) Show that if p(x) is a character of a group G then so is also 1/2{(p(x))^2 - p(x^2)}

(2) Let G be a finite group, A is a normal subgroup of G; suppose that for all a in A' = (A \ {1}), A = C_g (a) and that G\A is abelian.
Show that G has exactly (|A|-1)/|G:A| irreducible characters of degree greater than or equal to 2, which have degree |G:A|
• May 11th 2011, 06:34 PM
Drexel28
Quote:

Originally Posted by Turloughmack
Urgent answer required for these questions please!

(1) Show that if p(x) is a character of a group G then so is also 1/2{(p(x))^2 - p(x^2)}

Check Fulton and Harris this the natural representation on the exterior algebra ${\bigwedge}{}^2(V)$ (that is if $\rho:G\to\text{GL}(V)$ is the representation $\chi$ came from).

Quote:

(2) Let G be a finite group, A is a normal subgroup of G; suppose that for all a in A' = (A \ {1}), A = C_g (a) and that G\A is abelian.
Show that G has exactly (|A|-1)/|G:A| irreducible characters of degree greater than or equal to 2, which have degree |G:A|
You have yet to show any effort friend, come on.