# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### Character Table of the Monster

Does anyone have a link to a website having the character table of the Griess Monster in characteristic $0$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $g \in H$?
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### 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 ? ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Character table of $U_{16}$.

Find the character table of $U_{16}$. Could you give me a hint or a start? Thank you.
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### 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 ...
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### 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 ...