## The Index of the Center of a Character

**Point of post: **In this post we explore the index of the center of a character within a finite group.

*Motivation*

In our last post we saw how every character on a finite group naturally creates a host of subgroups of , namely the center of the character. We also saw how a lot of information pertaining to the index of in .

*Index of the Center of a Character*

* *

We discussed bef0re the notion of taking a representation and restricting it to a subgroup to create a representatino . Moreover, we discussed how the same metholody can take a character on and produce a character on by restricting to . Of course in general the representation and thus the the character that results from this restriction doesn’t have to be irreducible even if the original was (take any representation and ). But, there is a way in which one can bound the inner product of any character restricted to with itself by the value of the inner product of the original character with itself. Specifically:

**Theorem: ***Let be a finite group and . Moreover, let be a character on and the restriction of to . Then*

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* *

*where is meant to recall that we’re taking the inner product considering an element of the group algebra and the inner product on the group algebra . Moreover, equality holds if and only if vanishes on .*

**Proof: **One merely notes that

Moreover, the fact that equality holds if and only if vanishes on is also obvious from this.

From this e get the interesting corollary:

**Corollary:** *Let be a finite group and an irreducible character on . Then, .*

**Proof: **Recall from previous posts that for some degree-one irrep of . Thus, by first principles we know that and so by our previous theorem we get that

And this equality holds if and only if vanishes on .

In fact, we can say even more about when is abelian. Indeed:

**Theorem: ***Let be a finite group and a character on for which is abelian. Then .*

**Proof: **By our previous theorem it suffices to show that vanishes on . So, let . Now, we clearly must have that for some . Indeed, if this were true then we’d have that but since is an irrep we know from previous theorem that and so contrary to assumption. But, since is abelian we know that and so by the fundamental characterization of we have that (where is any representation which induces ). But, since we have that . Note though that and so . But and since is a class function this implies that . Thus, and since this implies that . The conclusion follows.

**References:**

1. Isaacs, I. Martin. *Character Theory of Finite Groups*. New York: Academic, 1976. Print.

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