All Questions
Tagged with characters finite-groups
77 questions
3
votes
1
answer
353
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
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
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
$$
1 = \sum_{\phi \in {\rm Irr}(G)} \frac{(d_\phi)^...
- 25.8k
3
votes
0
answers
400
views
Character table of the symmetric group $S_n$ according to James
In James, "The representation theory of the symmetric groups" an algorithm is described to produce the character table of a symmetric group. The proof involves the equation (pp. 22,23)
$$\...
- 2,050
3
votes
0
answers
75
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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
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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
3
votes
0
answers
113
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Length *vs* table of characters
Let $G$ be a finite group. Its length $\ell(G)$ is the length $n$ of any Jordan-Hölder filtration $1=G_n\subset\cdots\subset G_0=G$ ($G_i$ is normal in $G_{i-1}$ and $G_{i-1}/G_i$ is simple).
Can $\...
- 52.3k
3
votes
0
answers
441
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Degrees of irreducible characters of groups of order 48 [closed]
Hello
i wonder if there is a simple argument to show that no group of order 48 can have an irreducible character of degree larger than 4?
Thanks, Karim
- 41
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
526
views
Some special characters of finite groups
Let $G$ be a finite group, for each irreducible character $\chi$, we define ${\bf Z}(\chi)$ to be the set of all $x\in G$ such that $|\chi(x)|=\chi(e)$ when $e$ is the identity of the group.
For every ...
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
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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
88
views
When does a finite group have a lower-dimensional representation than one of its quotients?
The minimal dimension of a nontrivial representation of a central extension of a group $G$ can be smaller than the minimal dimension of a nontrivial representation of $G$. What about non-central ...
- 21
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
2
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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)...
1
vote
1
answer
181
views
Groups with many vanishing elements
It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is
What can be said about a finite group $G$ for which $\...
- 1,651
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
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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
1
vote
0
answers
82
views
associativity of the extension of finie groups [closed]
Following my previous question I have two questions:
1.the extensions of the group is associative or not, i.e. as we know by the notations of atlas if $S={\rm PSL}(2,q)$, where $q=p^f>9$, then $2....
- 53
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
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
0
votes
0
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112
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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{-...
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