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2
votes
3answers
392 views

a question about finite simple non-abelian groups

Is this true? Let $G\neq A_5$ be a finite simple non-abelian group. Then $G$ has a cyclic subgroup of order $2p$ and a subgroup isomorphic to the dihedral group of order $2p$, for some prime $p$.
4
votes
2answers
437 views

Transitive subgroup of $S_p$ containing a $p$-cycle and a double transposition

Let $p$ be a prime other than 5 or 7. Are $A_p$ and $S_p$ the only subgroups of $S_p$ that contains a $p$-cycle and a double transposition? As for $p = 5$, the dihedral group $D_{10}$ contains a ...
3
votes
1answer
158 views

Groups with special automorphism group

I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to $H$ is $\sigma$. Is ...
1
vote
0answers
136 views

Conjugacy classes of centralizers of semisimple elements in a finite group of Lie type

Let $G$ be a finite group of Lie type. By Deriziotis' and Carter's articles we know that conjugacy classes of connected centralizers of semisimple elements are parametrized by $(J,[w])$ where $J$ is a ...
3
votes
2answers
299 views

On finite groups with same complex-valued character table

What are the necessary and sufficient conditions for two finite groups $G$ and $H$ to have same complex-valued character table? Is there any criterion for which one could know about the character ...
25
votes
2answers
1k views

Why are Schur multipliers of finite simple groups so small?

Given a finite simple group $G$, we can consider the quasisimple extensions $\tilde G$ of $G$, that is to say central extensions which remain perfect. Some basic group cohomology (based on the ...
7
votes
2answers
314 views

Good effective versions of theorems of Artin and Brauer

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups. For example, Artin's theorem is the statement that for every character $\chi$ of ...
0
votes
0answers
111 views

How to find quotients of infinite triangle groups or von Dyck groups?

I need the following information about the quotients of infinite triangle (or von Dyck) groups. (1) Let $G(l,m,n)$ defined as $S^l$ =$T^m$ = $(ST)^n$ = $E$ is the hyperbolic ($1/l+1/m+1/n<1$) ...
3
votes
1answer
335 views

Special automorphisms of extraspecial groups

Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ consisting of all ...
4
votes
4answers
509 views

A catalog of faithful representations of finite groups?

I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$. Where ...
3
votes
2answers
319 views

group generated by Coxeter elements

Let $G$ a connected semisimple simply connected group over $\mathbb{C}$ and $W$ his Weyl group. What can be said about $W'$, the subgroup of $W$ generated by the Coxeter elements of $W$?
0
votes
0answers
103 views

Odd-order groups with homocyclic sylow subgroups

We say that $G$ is a homocyclic group, if it is direct product of isomorphic cyclic groups. Is there any classification of finite odd-order groups which all their Sylow subgroups are homocyclic?
3
votes
1answer
190 views

Relationship between the number of Sylow subgroups with element orders in finite group

In a finite group what is relationship between the number of Sylow $p$-subgroups with the number of elements of order a multiple of $p$? Is there any reference for my question?
4
votes
0answers
240 views

nilpotent fixed-point-free groups of automorphisms

Let $G$ be a finite group and $H$ be a nilpotent subgroup of $Aut(G)$. If $C_{G}(H)=1$, is $G$ solvable?
3
votes
2answers
347 views

element of order n such that $\pi(n)=\pi(G)$, where $\pi(n)$ denote the prime divisors of $n$

Hello. I thank for your answer, in advance. Let $G$ be a finite group and $G$ has an element of order $n$ such that $\pi(n)=\pi(G)$ where $\pi(n)$ denote the set of prime divisors of $n$ and $\pi(G)$ ...
0
votes
0answers
55 views

Computing the number of elements of order $2$ and $3$, in the groups $L_{3}(q)$

What are the number of elements of order $2$ and $3$ in the groups $L_{3}(q)$? Also let $r$ be a divisor of $q^{2}+q+1$. What is the number of elements of order $r$ in the groups $L_{3}(q)$?
10
votes
2answers
547 views

Embeddings of finite groups into GL(n,Q_p)

This question is inspired by some interesting comments on this recent question. Fix an integer $n \geq 1$ and a finite subgroup $G$ of $\mathrm{GL}_n(\mathbf{C})$. It is known that there are ...
3
votes
4answers
328 views

Intersection of all normalizers

This is probably standard for group-theorists: Let $G$ be a finite group. Is it true that the intersection of all normalizers of subgroups equals the center? If so, where do I find a proof? What about ...
0
votes
1answer
92 views

The automorphisms of a 2-group of nilpotency class 2

Let $p$ be a Merssene prime, i.e. $p=2^a-1$, where $a$ is a prime. Let $R$ be a 2-group of order $2(p+1)=2^{a+1}$. Also we know that $|Z(R)|=2$ and $R/Z(R)$ is abelian. Can we conclude that $R$ has ...
4
votes
1answer
230 views

Finite Unipotent Groups: References

It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}_q)$. (Thanks in advance!!!) 1) The number of ...
10
votes
2answers
409 views

Finite subgroups of $PGL(3,K)$

It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
1
vote
0answers
142 views

How many subgroups of order $\prod_{1}^{n} p_{i}^{n_{i}}$ are there in the finite product of cyclic groups?

All of the following ${p_{i},q_{i}}$are prime numbers, ${n,m,k}$ are pre-assigned integers. Consider the product of cyclic groups $\prod_{1}^{n} \mathbb Z_{p_{i}^{n_{i}}}$ then we asked the question: ...
0
votes
3answers
465 views

a group with all sylow p subgroups cyclic

If there exist a non cyclic group $G$ with all sylow $p$subgroups cyclic,and the normal $p_1$-complement $M$ for $G$ is cyclic,here $p_1$ is the smallest factor of $|G|$?And when does it always exist? ...
18
votes
6answers
1k views

Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements. I would like a measure that intuitively captures the extent to which $G$ is non-commutative. One easy measure is a count of the non-commutative ...
0
votes
1answer
89 views

lenght of a finite group versus number of conjugacy classes of subgroups

Let $G$ be a finite group. A chain of subgroups of $G$ of length $d$ is a sequence of subgroups of the form $$ \{e\}=G_0 \subsetneq G_1 \subsetneq \ldots \subsetneq G_{d-1} \subsetneq G_d=G. $$ ...
1
vote
1answer
489 views

A question on automorphisms of finite abelian groups

Which are the finite groups $(G,\cdot)$ with the following property: for every $f \in Aut(G)$, there are $g,h\in Aut(G)$ such that $f(x)=g(x)\cdot h(x), \forall x\in G?$ I already have verified ...
1
vote
1answer
217 views

Char $p$ representations of $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$

It is a well known fact that the irreducible representations of $SL_2(\mathbb{F}_p)$ over $\overline{\mathbb{F_p}}$ are given by the symmetric powers $Symm^k(V)$, where $V = ...
3
votes
2answers
249 views

Subgroups of $SL_2(F)$ generated by unipotent elements

I am interested in the following problem : given a finite field $F$ and two unipotent elements $g_1,g_2\in\mathrm{SL}_2(F)$ which do not commute, what can we say about the subgroup they generate? More ...
5
votes
0answers
101 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 ...
0
votes
1answer
204 views

Order of difference of two generators of cyclic group

Let $n\in\mathbb{Z}^+$ and $\alpha,\beta $ be two generators of the cyclic group $\left(\frac{\mathbb{Z}}{(2^n - 1)\mathbb{Z}},+\right)$. Question: What are known theorems regarding the order of ...
6
votes
2answers
331 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 ...
2
votes
1answer
188 views

Classification of generously transitive groups

A permutation group $G \lt S_n$ is called generously transitive, if for each $i,j$ there exists a permutation that interchanges them. Is there a reasonable classification of such (finite) groups?
6
votes
1answer
192 views

Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module

Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...
8
votes
2answers
333 views

Decomposition of induced representations / Refinement of Mackey's criterion

There are already some questions with almost the same title, but they are more restrictive. Let $G$ be a finite group, $H$ a subgroup, $V$ an irreducible representation of $H$, and $W=Ind_H^G V$ the ...
20
votes
1answer
731 views

Richness of the subgroup structure of p-groups

Given a prime $p$ and $n \in \mathbb{N}$, let $f_p(n)$ be the smallest number such that there is a group of order $p^{f_p(n)}$ which all groups of order $p^n$ embed into. What is the asymptotic growth ...
0
votes
0answers
28 views

Minimum number of solutions in a system of equalities and non-equalities

Let $k<N$ and $P_1, ..., P_{2k+1}, \lambda_1, ..., \lambda_k$ be elements of a finite group $G$ of size $N$. Find the minimum number of solution of the system $$P_{2i} + P_{2i+1} = \lambda_i, ...
4
votes
1answer
282 views

Finite Quotients of Free Groups

It is interesting FACT that given $l,m,n\geq 2$, there is (are) a finite group with elements $a,b$ such that $o(a)=l, o(b)=m$, and $o(ab)=n$ (see link for a nice example by Derek Holt / B. Sury). ...
1
vote
1answer
215 views

Bounds for conjugacy classes of subgroups

My question is rather general and the reason for this is that I am primarily interested in finding sources where I can read more about this type of problems. So here goes: To begin with, I want to ...
6
votes
0answers
187 views

Intersections of anisotropic tori with split Levi subgroups

Let G be a connected reductive group defined and split over a finite field k with Frobenius morphism F. Let T be an F-stable minisotropic maximal torus. Let P be an F-stable proper parabolic ...
2
votes
2answers
209 views

Asking about a quasicommutative semigroup

Honestly, I have been looking for an a finite Quasicommutative semigroup by surfing the web, but I could't. May I ask here to give me an example for such this kind of semigroup. I tried to built one ...
2
votes
1answer
164 views

Equivariant cohomology of finite group actions and invariant cohomology classes

Let $W$ be a finite group acting on a space $X$. In what generality is it true that $H^*_W(X) = H^* (X)^W$? We always have a map $H^*_W(X) \rightarrow H^* (X)^W$, but it is certainly not an ...
1
vote
2answers
246 views

When is PSU(2,q^2) = PSL(2,q) ?

The context for this question is from page 284 - 287 of Berger's paper: ...
1
vote
0answers
187 views

Interesting examples of minimal action on torus

Edit 1:This is a cross post on MSE. See math.stackexchange.com/q/289595/12952 Edit 2:I originally asked for finite group actions as I thought that will be easier. But as pointed out by Victor minimal ...
1
vote
0answers
237 views

Injective Mapping

Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. Elements of A and x can be complex numbers. How can i show if the mapping from x to y is injective for given A ...
4
votes
0answers
129 views

What is known about 2-modular representations of Ree groups of type $F_4$?

A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...
12
votes
2answers
587 views

Wedderburn's theorem for $\mathbb{Q}G$

Let $G$ be a finite group and let $\mathbb{Q}G=M_{n_1}(D_1)\times\cdots\times M_{n_k}(D_k)$ be the decomposition of $\mathbb{Q}G$ as a product of rings of matrices over divisions rings. Let $Z_i$ be ...
15
votes
1answer
553 views

Lower bounds on the number of elements in Sylow subgroups

I posted this question on Math.SE (link), but it didn't get any answers so I'm going to ask here. This is an edited version of the question. Let $p$ be a prime and $n \geq 1$ some integer. ...
7
votes
1answer
461 views

An extension of the converse to Hall's theorem.

This is an extension of this MSE question, in which I asked whether there was a counterexample to the following statement, Conjecture. If a finite group $G$ contains a $\lbrace p,q \rbrace$-Hall ...
1
vote
1answer
384 views

sylow subgroups of GL(n,q)

Dear all I am interested to know about the r-sylow subgroups of GL(n,q), is there any work about the structure of this subgroups?
1
vote
0answers
121 views

Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q

Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the ...