All Questions
47 questions
52
votes
0
answers
1k
views
Class function counting solutions of equation in finite group: when is it a virtual character?
Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by
$$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid w(x_1,\dots,...
- 6,919
42
votes
3
answers
3k
views
Are there "real" vs. "quaternionic" conjugacy classes in finite groups?
The complex irreps of a finite group come in three types: self-dual by a
symmetric form, self-dual by a symplectic form, and not self-dual at all.
In the first two cases, the character is real-valued, ...
- 27.8k
37
votes
0
answers
1k
views
Is this generalized character always a character?
Let $G$ be a finite group, and $p$ be a prime. Then there is a generalized character $\Psi$ of $G$ which takes value $0$ on all elements of order divisible by $p$, and has $\Psi(y)$ equal to the ...
- 44.4k
21
votes
0
answers
473
views
Is there a "direct" proof of the Galois symmetry on centre of group algebra?
Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$
This induces a linear ...
- 1,949
20
votes
3
answers
940
views
What did Frobenius prove about $M_{12}$?
I am interested in this paper which I can't read because it's in German:
Frobenius, G., Über die Charaktere der mehrfach transitiven Gruppen., Berl. Ber. 1904, 558-571 (1904). ZBL35.0154.02.
A free ...
- 11.2k
20
votes
2
answers
948
views
The finite groups with a zero entry in each column of its character table (except the first one)
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
17
votes
2
answers
860
views
The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group
Throughout $G$ is a finite, non-abelian group.
$\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$
Let $\Irr(G)$ be the set of ...
- 25.8k
12
votes
2
answers
926
views
Finite groups with integral character table
The character table of a finite group will be called integral if all its entries are integers. There are $11$ such groups up to order $16$, namely $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
11
votes
1
answer
218
views
A question about the adjoint of the Adams operations on representation rings
Let $G$ be a finite group, and $R(G)$ its representation ring over $\mathbb{C}$. We have the Adams operations $\psi^k:R(G)\rightarrow R(G)$, given on the level of characters by: $$\chi_{\psi^k{V}}(g)=\...
- 1,949
9
votes
2
answers
1k
views
Character table entries and sums of roots of unity
It is well-known that the entries of the character table of a finite group are sums of roots of unity.
Question: Is the converse true? Explicitly, given $z\in \mathbb{Z}[\mu_\infty]$, can I find a ...
- 1,261
9
votes
2
answers
1k
views
Irreducible reps and characters of $G \rtimes A$
Is there a theorem which classifies irreducible representations of semi-direct product of finite groups $G \rtimes A$, where $A$ is a finite abelian group and hence write down the character table for $...
user35360
9
votes
2
answers
525
views
Characterization of the family of simple groups PSL(2,q) by tensor multiplicity
Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$
Let the ...
8
votes
3
answers
1k
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 ...
- 17.6k
8
votes
1
answer
392
views
Is a $G$-invariant character $\theta$ of $H$ extendible to $G$?
Let $G/H\cong PSL(2,11)$, and $\theta$ be an irreducible $\mathbb{C}$-character of $H$. Suppose $\theta$ is invariant in $G$ and $\theta(1)=9$.
Question: Is $\theta$ extendible to $G$?
- 577
7
votes
1
answer
647
views
On the structure of a finite group of order $144$
Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$.
QUESTION: Prove $G\cong A_4\times A_4$.
By using Magma, we know ...
- 577
7
votes
1
answer
322
views
Why is Nagao's theorem the "Module theoretic version of Brauer's second main theorem"?
Let $G$ be a finite group, $p\in\mathbb{P}$ a prime, $\mathbb{F}$ an algebraically closed field of characteristic $p$, and $D\leq G$ a $p$-subgroup.
Brauer second main theorem states
If $\chi\in ...
- 9,650
7
votes
1
answer
197
views
A class function defined using Frobenius-Schur indicators
Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in ...
- 1,090
7
votes
0
answers
206
views
Rings that are $K_0$ of finite groups
Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...
- 2,934
6
votes
1
answer
298
views
Is a finite group given by its character table if its Sylow subgroups are so?
As pointed out by Mikko Korhonen in this answer, Özdem Çelik proved (in 1976 here) that a finite group whose Sylow subgroups are cyclic (called a Z-group) is determined by its character table.
...
6
votes
0
answers
320
views
(CFSG-free) Finite simple groups whose character degrees square divide its order
Let $G$ be a finite group. It is well-known that for all irreducible complex character $\chi$ then $\deg(\chi)$ divides $\lvert G\rvert$. Motivated by some problems with modular tensor categories, we ...
5
votes
2
answers
348
views
$G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$?
Let $G$ be a finite group of order $240$.
If $G\cong C_4\times A_5$ or $C_2\times C_2\times A_5$, then the all degrees of irreducible $\mathbb{C}$-characters of $G$ are
$
[1,1,1,1,~3,3,3,3,3,3,3,3, ~...
- 577
5
votes
1
answer
216
views
To whom is the internal characterization of $Q$-groups due?
A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following:
$G$ is a $Q$-group if and ...
- 5,654
5
votes
0
answers
991
views
Character table of a group determines the set of commutators of the group
We write $[x,y]$ for the commutator $x^{-1}y^{-1}xy$ of $x$ and $y$ in a group $G$.
(A) Let $g \in G$ and fix $x \in G$. Show that $g$ is conjugate to $[x,y]$ for some $y \in G$ iff
$$\sum_{\chi \...
- 428
4
votes
1
answer
518
views
Finite groups with the same character table *including* class types, and square-free order
There are non-isomorphic finite groups with the same (complex) character table, as $D_4$ and $Q_8$.
$$\scriptsize\begin{array}{c|c}
\text{class}&1&2A&2B&2C&4 \newline
\text{...
4
votes
1
answer
423
views
A global code for the character table of PSL(2,q)
We can easily get the character table of $\mathrm{PSL}(2,q)$ for some fixed small prime power $q$, we can just do (for example):
...
4
votes
1
answer
287
views
Character values of principal series representations of $GL_n(\mathbb{F}_q)$
Let $P_{\alpha}$ be the principal series representation of $GL_n(\mathbb{F}_q)$, where $\alpha = ( \alpha_1, \alpha_2, \cdots, \alpha_n)$ and $\alpha_i : \mathbb{F}_q^* \rightarrow \mathbb{C}^*$.
...
- 73
4
votes
0
answers
98
views
Is there a cohomological interpretation of the bilinear form arising from Clifford theory?
For this question all groups are finite, and representations are over $\mathbb{C}$. The setup is that we have $N$ a normal subgroup of $G$, with abelian quotient $A$, and an irrep $V$ of $N$ that is ...
- 1,949
4
votes
0
answers
227
views
Orbits of group representation over $\mathbb{F}_2$
Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
- 161
4
votes
0
answers
348
views
Interplay between two definitions of the transfer homomorphism.
The transfer homomorphism can be defined in a couple of ways. For the purposes of this question, assume that all groups are finite.
Defintion 1. Let $1\leqslant K'\leqslant L \leqslant K\leqslant G$ ...
- 1,173
3
votes
2
answers
222
views
Equality of subsets of abelian groups
Let $G$ be a finite abelian group, $X$ and $Y$ be two non-empty subsets of $G$ of equal size. Suppose that for each irreducible character $\chi$ of $G$ we have $\sum_{x\in X}\chi(x)=\sum_{y\in Y}\chi(...
- 105
3
votes
1
answer
321
views
Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?
Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation.
The corresponding Hurwitz representation $V_{\text{Hur}}$ has character $$\...
- 175
3
votes
1
answer
352
views
Computing characters of $\alpha$-projective representations
Given a finite group $G$, a finite cyclic group $A$ (viewed as a subgroup of $\mathbb{C}^{\times}$, i.e. generated by a $|A|$-th root of unity), and a 2-cocycle $\alpha\in Z^{2}(G,A)$. Recall that an $...
- 547
3
votes
0
answers
400
views
Character table of the symmetric group $S_n$ according to James
In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...
- 2,050
3
votes
0
answers
75
views
Expressing $\sum_{g\in [G/H]}ge_Hg^{-1}\in Z(\mathbb{C}[G])$ in terms of primitive central idempotents?
Suppose $G$ is a finite group, and $H$ a subgroup. For an irreducible character $\chi$ of $G$, there is a central idempotent in the group algebra $\mathbb{C}[G]$:
$$
e_\chi=\frac{\chi(1)}{|G|}\sum_{g\...
- 131
3
votes
0
answers
130
views
Does $G$ have a normal abelian Sylow $2$-subgroup?
Let $G$ be a finite group. Let $|G|=2^\alpha n$ where $(2,n)=1$ and $\alpha$ is a positive integer. Suppose that $\def\cd{\operatorname{cd}} n=\max \cd(G)$, and $n^2>\frac{1}{2}|G|$, where $\cd(G)$...
- 577
2
votes
1
answer
220
views
The lower bound of a group with characters of special degrees
Is there any lower bound for the order of a group with an irreducible character of degree $p$, where $p$ is a prime.
Is there any similar result for $p^2$ or $p^3$ instead of $p$?
Thanks for your ...
- 1,168
2
votes
1
answer
251
views
In which finite groups is there a non-central g such that, for all irreducible characters, Chi(i)(g) <> zero?
What is the character of Pi(G), the tensor product of all inequivalent irreducible representations of G?
- 49
2
votes
1
answer
130
views
Existence of universal bound related to characters
Let $G$ be a finite group and $\lambda \in \text{Irr}(G)$ an irreducible complex character of $G$.
Let $m(\lambda) := \min \{ \vert G : H \vert \mid H \leq G, \lambda\vert_H \text{ has a linear ...
- 83
2
votes
1
answer
112
views
Restriction of real irreducible 2-Brauer characters to subnormal subgroups
Question: Find a finite group $G$, a subnormal subgroup $H$ of $G$, a real-valued irreducible $2$-Brauer character $\chi$ of $G$ and a real-valued irreducible $2$-Brauer character $\mu$ of $H$ such ...
- 1,090
2
votes
0
answers
220
views
Characters of alternating groups
I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are:
A character of dimension $3.696$ of $A_{...
- 370
2
votes
0
answers
124
views
Action of diagonal automorphisms on the set of irreducible characters of $D_n(q)$
Let $S$ be $D_n(q)$ where $q$ is a prime power. We know that a diagonal automorphism $\phi_h$ of $S$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ normalizes $S$. Note that $\...
- 613
2
votes
0
answers
219
views
The tallest possible lattice?
Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...
- 298
1
vote
1
answer
270
views
On the character degrees of a finite group with special structure
Let $G$ be a finite group such that $G$ has a normal subgroup $N$ of order $p(p^2+1)/2$, where $p>13$ is an odd prime and $p\ne 239$. Also $G/N\cong \text{PSL}(2,p)$. Can we say that there exists a ...
- 1,168
1
vote
0
answers
179
views
Character extension about $Q_8$
Recently, I am studying the book Navarro - Character Theory and the McKay Conjecture. I am trying to solve the following exercise:
(Exercise 5.9)
Let $G$ be a finite group and $N\unlhd G$, suppose ...
- 195
1
vote
0
answers
103
views
Character degrees of a finite group?
Let $G$ be a finite group, $R(G)$ be the solvable radical of $G$, and $G/R(G)$ be an almost simple group with socle $S\cong PSL_2(3^p)$, where $p$ is an odd prime. Also let $[G/R(G):S]=p$ and $\theta$ ...
- 841
1
vote
0
answers
76
views
The degrees of ordinary characters of $PSp(2n,q)$ and $P\Omega O(2n+1,q)$
The finite simple groups $PSp(2n,q)$ and $P\Omega O(2n+1,q)$ have a same order, where $n\geqslant3$ and $q$ is odd.
What are the degrees of the ordinary characters of these two groups?
Thanks!!!
- 577
0
votes
1
answer
212
views
Is $G$ non-solvable?
Let $G$ be a finite group of order $2^7\cdot3^3\cdot5^2\cdot7$. Let $\mathrm{Irr}(G)$ be the set of all the irreducible $\mathbb{C}$-characters. Suppose that
(1) there is a character $\chi\in\mathrm{...
- 577