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5
votes
3answers
425 views

Exponent of Sylow $p$-subgroup of classical groups over a field of characteristic $p$

Let $G$ be a classical group of dimension $n$ over $GF(q)$ where $q=p^f$ is a prime power, and $P$ be a Sylow $p$-subgroup of $G$. What is the maximal order of elements, i.e. the exponent, of $P$? ...
2
votes
2answers
331 views

finite groups with trivial frattini subgroup

Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and atleast one of its ...
2
votes
1answer
157 views

maximal subgroups of finite simple groups

Is it possible to classify finite simple groups whose every maximal subgroups are not of prime order? Is it possible to answer to this question in the class of finite groups?
8
votes
2answers
384 views

A conjecture on solvablity of finite groups

Suppose $G$ is a finite group and $A$ an abelian subgroup. Suppose for some natural number $n\geq 2$, elements of $\gamma_n(G)$ have the form $[a, x]$ where $a\in A$ and $x\in G$. Then $G$ is ...
0
votes
1answer
167 views

Prime divisor of finite group

We know that the number of elements of order $k$ in a finite group $G$ is equal to $\sum |cl_{G}(x_{i})|$=$\sum|G/C_{G}(x_{i})|$ such that $|x_{i}|=k$. It is clear that for a prime $p$ if $p\mid ...
3
votes
0answers
83 views

On divisors occurring as subgroup sizes

Given a finite group $G$ define $D(G)$ to be the number of divisors $r$ of $|G|$ for which there exists a subgroup of $G$ of order $r$. Clearly $D(G) \leq d(|G|)$, where $d(n)$ denotes the number of ...
8
votes
1answer
391 views

Groups with an automorphism of order two fixing only two elements

It is well known that a finite group admitting an automorphism of order 2 that fixes only the identity is abelian and has odd order. Moreover, the automorphism is inversion. Is anything known about ...
1
vote
1answer
181 views

How we characterize a subgroup of finite group of Lie type with unipotent elements.

Let $G$ be a finite group of Lie type. Let $H$ be a subgroup of $G$ which contains unipotent elements. I want to find a 'nice' subgroup of $G$ that contains $H$, for example a Levi subgroup of $G$ ...
3
votes
2answers
280 views

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ stabilizing an isotropic subspace. Is the Borel subgroup of $G$ maximal soluble in $P$ and is there ...
7
votes
2answers
400 views

Automorphism Group of a p-group : Looking for a Reference

In the following post by DavidLHarden : See Here He quoted the following claim: "There is a theorem that says that if $p$ is a prime and $|G|=p^n $ , then $|AutG| $ divides $ \Pi_{k=0}^{n-1} ...
4
votes
7answers
694 views

maximal subgroups of finite simple groups

Is it possible to determine the structure of maximal subgroups of finite simple groups?(Even if in special cases such as minimal simple groups, alternating groups,...)
4
votes
2answers
470 views

solvable groups

Let $G$ be a finite group which has a cyclic maximal subgroup. Is $G$ solvable?
3
votes
0answers
232 views

How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?

This is a crosspost from MSE since I haven't found an answer there yet. I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use ...
0
votes
1answer
121 views

sylow basis of finite solvable groups

Is it possible that an element of a sylow basis of a finite solvable group G lies in the frattini subgroup of G?(i.e could an element of a sylow basis of a finite solvable group G be a non-generator ...
2
votes
1answer
215 views

Complexity of establishing finite groups (non)-isomorphism ?

Question Given two finite groups G and H of the same order N what are the algorithms and what is their complexity (in terms of N) to check is G isomorphic to H or not ? Is there polynomial in N ...
8
votes
3answers
1k views

Algorithm to check is representation irreducible ? Algorithm to decompose the reducible one ?

Question 1 Given a representation of a finite group, what algorithm can be used to check is it irreducible or not ? (Main case - complex numbers, comments on other cases are also welcome. "Given" ...
7
votes
2answers
576 views

How to compute all irreducible representations of a finite group ? (how GAP is doing this?)

Let us "take" a finite group G. Here "take" I mean any type of group-theoretic description you prefer: e.g. as an explicit subset of GL (or other group) or Cayley table, whatever. Question: How ...
0
votes
0answers
128 views

Finite reflection groups

Hello, Let $V$ an euclidean space, $X$ a finite set of non-zero vertors in V and $\mathcal{H}$ be the set of hyperplanes of the form $a^{\perp}$ for some $a\in X$. Let $W$ be the group generated by ...
4
votes
5answers
602 views

Product of conjugacy classes - is there an analog of Tanaka-Krein reconstruction ?

Consider a finite group G. The product of conjugacy classes can be defined in natural way just by multiplying the representatives and counting multiplicities (see e.g. MO 62088). So we get ring with ...
3
votes
1answer
275 views

Character theory of $2$-Frobenius groups.

This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there. Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...
6
votes
0answers
233 views

Recovering a group from its number of symmetric embeddings?

Fix a finite group $G$, and let $f_G:\mathbb{N}\rightarrow\mathbb{N}$ be defined by setting $f_G(n)$ to be the largest $m$ such that $S_n$ contains $m$ disjoint (pairwise-intersecting at 1) copies of ...
8
votes
2answers
518 views

Automorphisms of subgroup of hamming cube under distance constraint

Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of $S$. There's a ...
6
votes
0answers
311 views

Cyclic Sylow $p$-subgroup in finite simple groups

Hello, I came accross a beautiful and "simple" (no pun intended) theorem, mentioned here in slide 14 by Jack Schmidt. All finite simple groups have a cyclic Sylow $p$-subgroup for some $p$ I ...
8
votes
3answers
531 views

The powers of non-empty subset of a group that generate a subgroup

If G is a group and A and B to non-empty subsets of G, then by AB we mean the set consist of all product ab where a is in A and b is in B.(Standard definition) Similarly we can define X^m where X is a ...
0
votes
0answers
77 views

Building an invariant Sn structure from two invariant Zn structures

Take two mathematical structures with a $Z_n$ symmetry (cyclic symmetry). Which are the different ways, in "gluing" these structures, to obtain a mathematical structure with a $S_n$ symmetry ...
4
votes
0answers
286 views

Finite groups that admit an anti-automorphism with many fixed points [duplicate]

Possible Duplicate: Homomorphism more than 3/4 the inverse Let $G$ be a finite group that admits an anti-automorphism $u: G \to G$ and let $S$ be the set of fixed points of $u$. I am ...
2
votes
2answers
279 views

If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ?

Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real"(symmetric) ? And vice verse ? ...
18
votes
5answers
1k views

Applications of Frobenius theorem and conjecture

A theorem of Frobenius states that if $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Frobenius conjectured that if the number of ...
3
votes
2answers
343 views

The cycle structure of twisted wires, connected

Suppose you have $n$ (blue) wires linearly arrayed at junction box $A$, connected to a remote junction box $B$, where the wires are now arrayed along a line in a randomly permuted order, i.e., each ...
12
votes
4answers
1k views

Has any attempt been made to classify finite groupoids?

I recently stumbled upon the Mathieu groupoid and I found them fascinating. It appears as a subset of $S_{13}$ which is not closed under multiplication, but it turns out to be a groupoid with 13 ...
2
votes
1answer
490 views

Linear algebra of finite abelian groups

If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...
7
votes
0answers
214 views

Why would dim primitive irrep divide size of some conjugacy class ?

From Isaacs et.al. 2005 Conjecture C. Let χ be a primitive irreducible character of an arbitrary finite group G. Then χ(1) divides | clG(g)| for some element g ∈ G. Here, of course, we ...
1
vote
0answers
127 views

What are natural automorphisms of set of subsets ? How to “constructify” Andreas Blass theorem on sets M,N with G action such that C[M] = C[N] as G modules ?

Consider vector space V over finite field $F_q$ and $V^ * $ its dual space. Denote $P(V), P(V^ * )$ the sets of ALL subsets in $V$ and $V^*$. Question How to construct GL_n(F_q) equivariant ...
5
votes
0answers
250 views

Vanishing sums of powers modulo n

Is it possible to describe all positive integer sequences $\{x_i\}$ such that $$ \sum_i x_i=n \quad \hbox{and} \quad \sum_i x_i^k\equiv 0\pmod n \quad \hbox{for all $k$ (and a given $n$)?} $$ ...
15
votes
1answer
624 views

Number of 2-dimensional irreducible representations of a finite group ?

Question: What is the number of two-dimensional irreducible representations of a finite group ? How it can be expressed in groups-theoretic terms ? (Number of 1-dimensional irreps is |G/[G,G]| ). ...
3
votes
1answer
263 views

On the existence of a direct summand containing a fixed subgroup

Let $G$ be a finite abelian group, and $g_1, \ldots, g_n \in G$ such that the cyclic groups that they generate are in direct sum $\langle g_1 \rangle \oplus \ldots \oplus \langle g_n \rangle$. Is it ...
13
votes
1answer
680 views

Non-isomorphic finite simple groups

Hello, The smallest integer $n$ such that there exists two non-isomorphic simple groups of order $n$, is $n=20160$ (namely for the groups $\mathrm{PSL}_3(\mathbb F _4)$ and $\mathrm{PSL}_4(\mathbb F ...
7
votes
2answers
398 views

Regular elements in the torus of a group of Lie type

Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...
11
votes
4answers
695 views

Realizable Order Sequences for Finite Groups

My post is motivated at least in part by this MO question. Has there been any work done on realizable order sequences for finite groups? By an "order sequence" I mean a non-decreasing list of the ...
1
vote
1answer
202 views

sum_g g^k, Frobenius-Schur indicators, S_n-invariants in freeAss(x_i), center of the group algebra

For any $k$ sums $T_k = 1/|G|\sum_{g\in G} g^k$ belong to the center of the group algebra, for finite group G. For $k=2$ they "are" (up to details and interpretation) Frobenius-Schur indicators. For ...
3
votes
4answers
231 views

Sets M,N with G action such that C[M] = C[N] as G modules, how are they related ?

Consider two sets M, N such that C[M] is isomorphic to C[N] as representations of G, somewhat surprisingly it does not imply M, N are isomorphic as G sets. (Everything is finite.) However it does ...
6
votes
3answers
472 views

Center and representations of finite group - how are related ?

If finite group G has a center how does it influence the representations of this group ? And vice versa - can we see somehow the center (or some of its properties) from representations (from ...
3
votes
3answers
444 views

Representation theory of p-groups in particular upper tringular matrices over F_p

Finite p-groups - have p^n elements by definition. According to WP there is rich structure theory. Question: How far is representation theory of p-groups is understood? In case this question is too ...
20
votes
0answers
324 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 ...
1
vote
0answers
218 views

Reference request

I am looking for a reference or proof for the following problem: Problem: Let $r$ be prime, then $2r$ is a Sylow $p$-number if and only if $2r=1+p^{2^n}$. Thanks in advance.
12
votes
3answers
1k views

Finite groups such that every irrep can be induced from trivial irrep of a subgroup ?

What are examples (general features) of the finite groups $G$, such that every irrep (irreducible representation) is contained (as constituent) in the representation induced from trivial ...
1
vote
0answers
280 views

about union of conjugate proper subgroups in a math paper

My question is about the shaded area in this image. Does the symbol $L=\bigcup_{g \in G} T^{g}$ means that $L$ is a union of sets or $L=\langle T^{g}, g\in G \rangle$? If it means the first one, then ...
16
votes
1answer
478 views

Ratio of number of subgroups to the order of a finite group

Let $\mathcal{G}$ be the set of finite groups and for $G \in \mathcal{G}$, let $S(G)$ be the set of subgroups of $G$. I am interested in the ratio $R(G)=|S(G)|/|G|$. It is easy to show that by picking ...
0
votes
1answer
185 views

What is a “non-splitting covering” of a finite group?

Apologies if this is elementary, but I have never heard the terminology before: What is a "non-splitting covering" of a finite group? I encountered the term while reading this paper, in which ...
14
votes
2answers
526 views

Maximal number of maximal subgroups

Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...