1
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2answers
59 views

Easy way to get real irreducible characters (reps) from complex irreducible characters?

For plenty of groups, the real irreducible characters/representations aren't the same as the complex irreducible representations. I really enjoy James Montaldi's summary of real representations, for ...
1
vote
1answer
48 views

How to tell whether a representation of a group is faithful or unfaithful?

From just the character table and the basis functions of the irreducible representations, how do I know whether a representation is faithful or unfaithful? For the 1-D representation it is trivial to ...
0
votes
0answers
25 views

Show that 2 representations are not equivalent and find all the irreducible representations of G.

Show that 2 representations are not equivalent and find all the irreducible representations of $G$. The group $G=T_{16}$ has order 16 and presentation given by $G=\langle a,b : a^8=b^2=1, ...
4
votes
0answers
20 views

Example of projective rep being used in Clifford theory

I'm trying to understand the use of projective representations in Clifford theory, and I'd like a small example where projective representations really help, and the ingredients are actually ...
2
votes
2answers
65 views

Nilpotent groups are monomial

I'm trying to show that a nilpotent group $G$ is monomial; i.e., that every irreducible representation $\rho$ of $G$ satisfies $\rho = \text{Ind}_H^G(\tau)$ for some $H \leq G$, $\tau$ a one ...
1
vote
1answer
43 views

Sum of irreducible character values in a row of the character table

If $\chi$ is a nontrivial irreducible character of $G$ (a finite group), define $S_{\chi}:= \sum_{x \in G} \chi(x)$. In terms of conjugacy classes $\mathcal{C}$, this is $\sum_{\mathcal{C}} ...
1
vote
1answer
39 views

A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
0
votes
1answer
110 views

Character Table of the Monster

Does anyone have a link to a website having the character table of the Griess Monster in characteristic $0$?
0
votes
0answers
23 views

subgroup of character group

Let $B$ be a subgroup of a finite Abelian group $A$ and $$B^0=\{\alpha \in \hat{A}\mid \alpha(b)=1 \text{ as } b \in B\}$$ Prove a. $B^0$ is subgroup of $\hat{A}.$ And for any subgroup ...
3
votes
1answer
60 views

Suzuki exceptional characters with $\epsilon = -1$

I have a question about Suzuki's theory of exceptional characters of finite groups. If you are familiar with this theory, then I'm just asking: can we always choose $\epsilon=1$? If not, here is a ...
1
vote
1answer
74 views

show that the following three statements are equivalent

Let $\Phi$ be an irreducible complex representation of the group $S_n$ and $\Phi'(\sigma)=\Phi(\sigma) \operatorname{sgn}(\sigma).$ $(\sigma \in S_n).$ Prove that 1) $\Phi'$ is ...
1
vote
2answers
60 views

Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$

Let $G$ be a non-trivial finite group. Prove that $\forall g \in G$ $ \exists$ an irreducible non-trivial character $\chi$ of the group $G$ such that $\chi(g)\neq 0$ This is my attempt so ...
1
vote
2answers
92 views

The average value of irreducible character of a non-trivial finite group

Let $G$ be a non-trivial finite group. Let $\chi$ be an irreducible character of the group $G$. Find $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}$$ I try. But I think that I am wrong. ...
3
votes
1answer
70 views

Irreducibility of complex 2-dimensional character of the group $ S_3 $

Let $\chi$ be 2-dimensional complex character of the group $ S_3 $. Prove that $\chi$ is irreducible character iff $\chi((123))=-1$. There is hint in my book: " Use the Maschke's theorem and ...
4
votes
1answer
71 views

Reading a Character Table in Magma

These are the outputs of two character tables from Magma. The first is for $A_5$. The second is for $GL_3(2)$. What is the significance of the '$+$' and '$0$' symbols? I can produce more tables if ...
1
vote
1answer
60 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
1
vote
1answer
40 views

If $G$ is solvable, is it true that for any $m,n\in\operatorname{cd}(G)$, there exists a prime $p$ such that $p\mid m,n$?

Let $G$ be a finite group and let $\operatorname{cd}(G)$ be the set of degrees of irreducible characters of $G$. It is known that if for any $m,n \in \operatorname{cd}(G) \setminus \{1\}$, there ...
2
votes
0answers
52 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
0
votes
0answers
35 views

a exercise in Berkovich‘ book

in Berkovich' book characters of finite groups there is a exercise in page 59. exercise 15, a group G is a Q-group if and only if for any cyclic subgroup Z of G who can tell me how to prove it ? ...
7
votes
1answer
138 views

character degree and solvability

There is an unsolved problem in Berkovich's book "Characters of Finite Groups Part 2" I state here: Is $G$ solvable if $\chi(1)^2$ divides $|G|$ for all $\chi \in \operatorname{Irr}{(G)}$? Can ...
4
votes
2answers
105 views

Computing bicharacters of (small) finite groups

I'm trying to find some finite groups with certain properites (hopefully of small order; no more than 100, I suspect), and one of the things I need to look at are all of its bilinear bicharacters: ...
1
vote
0answers
28 views

Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...
1
vote
2answers
152 views

Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$, where $N \unlhd G$ and $ \theta \in Irr(N)$.

Let $N \unlhd G$ and $ \theta \in Irr(N)$. Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$. Where $I_G(\theta)$ is the stabilizer of $\theta$ in the action of $G$ on $Irr(N)$ defined by ...
4
votes
3answers
319 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.

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
176 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
199 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 ...
5
votes
1answer
195 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
162 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 ...
2
votes
1answer
288 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
260 views

Character table of $U_{16}$.

Find the character table of $U_{16}$. Could you give me a hint or a start? Thank you.
11
votes
0answers
318 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 ...
1
vote
2answers
91 views

On a sum taken over all irreducible characters: is $\sum\limits_{i=1}^t\chi_{V_i}(g)^2$ nonzero for all $g\in G$?

Suppose $G$ is a finite group and $\mathbb{k}$ an algebraically closed field of characteristic zero. Denote the irreducible representations of $G$ over $\mathbb{k}$ by $V_1,\ldots, V_t$. Is it ...
6
votes
1answer
432 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. ...
8
votes
2answers
319 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 ...
7
votes
2answers
232 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 ...
2
votes
1answer
97 views

How to bound the order of a finite group under the following hypotheses?

In the book Character Theory Of Finite Groups by I.Martin Issacs as exercise 2.14 Let $G$ be a finite group with commutator subgroup $G'$. Let $H \subset G' \cap Z(G)$ be cyclic of order n and ...
17
votes
0answers
344 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
740 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 ...
11
votes
1answer
204 views

Do all algebraic integers in some $\mathbb{Z}[\zeta_n]$ occur among the character tables of finite groups?

The values of irreducible characters of a finite groups are always sums of roots of unity; do all sums of roots of unity (i.e. algebraic integers in the maximal abelian extension of $\mathbb{Q}$) ...
17
votes
2answers
937 views

exercise in Isaacs's book on Character Theory

I'm stuck on an exercise in Isaacs's book "Character Theory of Finite Groups" - it relates to something I'm looking at as part of ongoing research, but I guess it belongs here rather than on ...
8
votes
2answers
272 views

Functional equation of irreducible characters

I am preparing to an exam in representations of finite groups. I am trying to tackle a problem regarding a characterization of irreducible characters: Let $f$ be a complex-valued function on a finite ...