# Questions tagged [character-theory]

The character-theory tag has no usage guidance.

**2**

votes

**0**answers

44 views

### Properties of extendable irreducible characters to a normal Sylow subgroup

Let $G$ be a finite group with a nilpotent commutator subgroup, and denote by $mcd(G)$ the set of degrees of the irreducible monomial characters. Suppose $|mcd(g)|=2$. Furthermore, we call a monomial ...

**3**

votes

**1**answer

99 views

### Bounding the derived length of a solvable group given the degrees of the irreducible monomial characters

Much is known about the derived length of a solvable group given the degrees and cardinality of the set of degrees of the irreducible characters. Martin Isaacs and Donald Passman pretty much started ...

**1**

vote

**0**answers

31 views

### Irreducible characters of a semi-direct product with a p-group

Suppose G is a semi-direct product of P with H where P is a (non-abelian) p-group and G is solvable. I wonder what can be said about the irreducible characters of G given information about the ...

**1**

vote

**1**answer

92 views

### Study of the subgroups of which a non-linear monomial character is induced from

Let G be a monomial group, and let H_1,...,H_r be the subgroups of G where there exists a linear character that induces to an irreducible character of G. How much is known about these subgroups? For ...

**1**

vote

**0**answers

76 views

### If the kernel of an irreducible character contains the derived subgroup - is it then linear? [closed]

It is of course true that any linear character of a group contains the commutator subgroup. But is the converse also true? If not - do you know of a counterexample?

**1**

vote

**2**answers

171 views

### Character theory of representations of infinite groups

I saw that "Over an algebraically closed field of characteristic 0, semisimple representations are isomorphic if and only if they have the same character" in the Wikipedia page , which does not ...

**3**

votes

**0**answers

41 views

### Index of subgroup generated by characters induced from $p$-elementary subgroups in the ring of virtual characters

I posted this over on MSE, but received absolutely no love. So maybe I’ll have better luck here. It seems like a relatively easy group theory question that I’m just not seeing! It’s on the essential ...

**1**

vote

**0**answers

107 views

### We know $A_5$ as a non-CI-group. Now, is $A_5$ a BI-group?

We call a group satisfying the following property for all $\nu \in cd(G)$ (Irreducible character degrees of $G$) a BI-group (Babai Invariant group)
Let $G$ be a finite group, let $\Gamma=Cay(G,S)...

**8**

votes

**3**answers

762 views

### Beyond Brauer's theorem

Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary ...

**44**

votes

**5**answers

3k views

### How much of the ATLAS of finite groups is independently checked and/or computer verified?

In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ...

**17**

votes

**1**answer

521 views

### Number of solutions to equations in finite groups

Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$.
Is it always true that the number ...

**6**

votes

**2**answers

295 views

### Subgroups from which all class functions extend to class functions on the ambient group

Until someone suggests better terminology, let me call a subgroup H of a finite group G segregated if every class function on H can be extended to a class function on G. Equivalently, H should have ...

**2**

votes

**1**answer

333 views

### Is there any groups $G$ with the property $(*_d)$?

Let $G$ be a finite group of even order has only one non-principal irreducible character $\chi$ of degree $d$, $d\in \mathbb{N}$, with the following property (we name it $(*_d)$):
$(*_d)$: There ...

**1**

vote

**0**answers

148 views

### Do you know any clear classification of groups in which there would exist a unique non-linear character of a given degree?

According to
Lev Kazarin, On Thompson’s Theorem, Journal of Algebra 220, 574–590 (1999)
we know that:
[Corollary 5.3]:Let $$cd(G)=\{\chi(1)|\chi\in Irr(G)\}=\{1,f_1,\dots,f_n,d\}, \;\;n\gt0,$$
...

**6**

votes

**1**answer

257 views

### A constant associated to the character table of a finite group

Following on from some of myprevious MO questions on finite group theory...
$\newcommand{\Irr}{\operatorname{Irr}}\newcommand{\Conj}{\operatorname{Conj}}\newcommand{\AMZL}{{\rm AM}_{\rm Z}}$
Let $G$ ...

**2**

votes

**2**answers

346 views

### reference request for character theory of p-extraspecial groups

In a recent preprint 1302.1929 my co-authors and I make use of some results about the character theory of extraspecial groups. These were largely gleaned from haphazard Googling, comments made by ...

**2**

votes

**0**answers

147 views

### The largest number of irreducible characters of the same degree in a finite group

Dear all,
For a finite group $G$, let $m(G)$ denote the largest number of irreducible characters of the same degree of $G$. You can say that $m(G)$ is the largest multiplicity of character degrees of ...

**5**

votes

**0**answers

160 views

### Information about permutation character from local action

Let $G$ be a finite permutation group acting transitively, but not regularly, on a set $V$. Let $H$ be the stabilizer of some point $v\in V$, and suppose that $H$ acts 2-transitively on one of its (...