Tagged Questions

10
votes
5answers
189 views

Applications of Character Theory

Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
2
votes
2answers
101 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.

Suppose $G$ is an abelian group and $a\in G$ and $$f:\left<a \right>\to\Bbb T$$ is a homomorphism. Can $f$ be extended to a homomorphism on $G$: $$g:G\to \Bbb T$$ ? $\Bbb T$ is the circle ...
3
votes
1answer
65 views

Exercise 2.15 M.Isaacs' Character theory of finite groups

I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book. I would need some help with this one: (2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
1
vote
1answer
70 views

Exercise 2.8 M.Isaacs' Character theory of finite groups

I'm a starter at character theory. I'm trying to do this exercise: (2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible ...
4
votes
1answer
81 views

Character theory exercises [closed]

I'm doing the exercises from chapter 2 of M.Isaacs' Character theory of finite groups, and I'm having problems with some of them. In particular, I would need help with these ones. Thank you very much ...
1
vote
1answer
90 views

Irreducible Representations and Maschke's Theorem

$\mathscr L(V,W)^G = \mathscr L(V_1,W)^G \oplus...\oplus\mathscr L(V_k,W)^G$ and for each irreducible represtation of G on a space W, the number of $j\in (1,...,k)$ for which $V_j \cong W$ is ...
1
vote
0answers
36 views

Isomorphism of annihilator of a subgroup in the context of group characters

I am trying to learn about characters of finite abelian groups. A character is a homomorphism from a finite abelian group $G$ into the multiplicative group of complex numbers of absolute value 1. In ...
2
votes
1answer
69 views

Sum of squares of dimensions of irreducible characters.

For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...
0
votes
1answer
214 views

Character table of $U_{16}$.

Find the character table of $U_{16}$. Could you give me a hint or a start? Thank you.
8
votes
0answers
200 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...
3
votes
3answers
92 views

Character of $S_3$

I am trying to learn about the characters of a group but I think I am missing something. Consider $S_3$. This has three elements which fix one thing, two elements which fix nothing and one element ...
6
votes
1answer
167 views

What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. ...
5
votes
2answers
214 views

What is an irrreducible character of a finite group?

Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b ...
5
votes
2answers
118 views

Estimates on conjugacy classes of a finite group.

In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32. Theorem: Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
4
votes
2answers
155 views

Two non-isomorphic groups with the same complex character table

Could you give me an example of two non-isomorphic groups with the same complex character table?
10
votes
0answers
226 views

Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In ...
3
votes
2answers
328 views

Condition for abelian subgroup to be normal

Sorry for any mistakes I make here, this is my first post here. I have a group $G$ which has an abelian subgroup $A<G$. I also know there is a irreducible character $\chi$ with the degree of ...
7
votes
3answers
356 views

Character Table From Presentation

I've recently learned about character tables, and some of the tricks for computing them for finite groups (quals...) but I've been having problems actually doing it. Thus, my question is (A) how to ...
2
votes
1answer
207 views

Setting up Brauer character theory

My question relates to p. 147 of Serre's Linear Representations of Finite Groups, where he is setting up the definitions relevant to Brauer character theory. Having fixed an algebraically closed ...
3
votes
1answer
128 views

The natural inclusion of an infinite abelian group $G$ into $\widehat{\widehat{G}}$

I was recently trying to think of a simple example that demonstrates that the natural inclusion of an abelian group $G$ into ...