All Questions
Tagged with characters finite-groups
77 questions
8
votes
0
answers
190
views
Groups having exactly two non real-valued irreducible characters
This is an enlarged version of my question on MSE. It was suggested I ask here instead.
Suppose the finite group $G$ has exactly two conjugacy classes that are not self-inverse (a conjugacy class is ...
- 787
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
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 $$\...
- 175
17
votes
1
answer
1k
views
Explicit character tables of non-existent finite simple groups
In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...
3
votes
0
answers
172
views
Abelian characters and odd perfect numbers?
This question is about applications of abelian characters to odd perfect numbers:
Context and Definitions:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this set to a ring ...
- 3,064
5
votes
1
answer
342
views
Product of all conjugacy classes
Related to this post of my coauthor Sebastien, I should also mention that one can also prove the following dual result:
For any finite group G, the following identity holds:
$$
\left(\prod_{j=0}^m \...
- 887
5
votes
0
answers
351
views
Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
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
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
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 ...
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$, $...
7
votes
0
answers
139
views
When is $\operatorname{Out}(G)$ isomorphic to the symmetries of the character table of $G$?
$\DeclareMathOperator\Out{Out}\DeclareMathOperator\CTS{CTS}$The outer automorphism group $\Out(G)$ of a finite group $G$ act on the character table by permuting columns (conjugacy classes) and rows (...
- 141
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
3
votes
0
answers
102
views
Terminology for the "natural probability measure" on the set of irreducible characters of a finite group
To be specific: if $G$ is a finite group and $\operatorname{Irr}(G)$ the set of its irreducible characters (over the complex field) then we know that
$$
1 = \sum_{\phi \in {\rm Irr}(G)} \frac{(d_\phi)^...
- 25.8k
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
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
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
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
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
0
votes
0
answers
53
views
Counting the number of generating triples of various types in finite simple groups
I am trying to figure out how specific generating triples in finite simple groups are calculated. My understanding is that it uses Frobenius's formula and character theory. I'm not an expert on ...
- 29
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 \...
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
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
7
votes
2
answers
764
views
Proofs of a character identity?
Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C_1,...,C_k)$ a tuple of conjugacy classes of $G$. I am interested in proofs of the following identity
$$
\sum_{(c_1,...,c_k) \in C_1 \...
- 5,349
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 ...
4
votes
1
answer
196
views
Generalization of $\lim_{n \rightarrow \infty} \prod_{i=1}^{n}\frac{2i-1}{2i}$ for a character $\chi:\mathbb{Z}/s \mathbb{Z} \rightarrow \mathbb{C}^*$
Playing with some infinite products I came up with this problem, that I'm not able to figure it out by myself. Moreover in the internet it doesn't seem to appear anywhere.
Maybe it is just an easy ...
- 1,343
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):
...
1
vote
2
answers
411
views
Are the character degrees determined by the conjugacy class sizes?
The computation below (part 1) shows that if two finite groups of order at most $100$ have the same (ordered) list of conjugacy class sizes, then they also have the same (ordered) list of (irreducible)...
4
votes
0
answers
153
views
Characters, centralizers and cosets
I am trying to understand/count the number of solutions of a number of equations in finite groups and came across the following class function:
$$ \theta_\chi(x) = \sum_{y\in G} |C_G(xy)y \cap C_G(x)| ...
7
votes
3
answers
990
views
The zero entries in the character table of a finite group
When you browse the character tables of the small finite groups (for example here), you can observe that every zero entry corresponds to the value of an irreducible character $\chi$ on a non-central ...
3
votes
0
answers
400
views
Character table of the symmetric group $S_n$ according to James
In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...
- 2,050
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
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
0
votes
0
answers
112
views
Definition of "tame $p$-part of $\chi$"
At the beginning of Haruzo Hida's article "Big Galois representations and $p$-adic L functions", he has defined $$\chi_1= \textrm{the } N_0 \textrm{-part of} \chi \times \textrm{the tame } p\textrm{-...
- 337
11
votes
1
answer
550
views
Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$
$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$Let $G$ be the symmetric group $S_n$ or the projective general linear group $\PGL_2(n)$.
Let $X$ be a cyclically reduced word in the ...
- 9,501
3
votes
0
answers
75
views
Expressing $\sum_{g\in [G/H]}ge_Hg^{-1}\in Z(\mathbb{C}[G])$ in terms of primitive central idempotents?
Suppose $G$ is a finite group, and $H$ a subgroup. For an irreducible character $\chi$ of $G$, there is a central idempotent in the group algebra $\mathbb{C}[G]$:
$$
e_\chi=\frac{\chi(1)}{|G|}\sum_{g\...
- 131
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.
...
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{...
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
3
votes
1
answer
1k
views
Conditions for a solvable group to have a non-trivial center
I am working on a problem in character theory where I try to bound the derived length of a solvable group using information about its characters. In my specific case, it will be extremely helpful for ...
- 415
6
votes
1
answer
276
views
Reference request: an elementary result on characters of finite abelian groups
The referee of a paper I submitted to a journal asked me to include a reference of the following elementary result on characters of finite abelian groups:
Let $A$ be a finite abelian group of order $...
- 3,107
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
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
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
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
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
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
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
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
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