# Tagged Questions

The finite-groups tag has no wiki summary.

**25**

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**2**answers

997 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

**2**answers

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**

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**0**answers

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

**1**answer

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

**4**answers

508 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

**2**answers

317 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**

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**0**answers

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

**1**answer

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

**0**answers

239 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

**2**answers

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

**0**answers

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

**2**answers

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

**4**answers

327 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

**1**answer

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

**1**answer

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

**2**answers

407 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

**0**answers

141 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

**3**answers

461 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

**6**answers

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

**1**answer

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

**1**answer

487 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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

203 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

**2**answers

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

**1**answer

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**

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**1**answer

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

**2**answers

332 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

**1**answer

730 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

**0**answers

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

**1**answer

280 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

**1**answer

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**

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**0**answers

185 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**

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**2**answers

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

**1**answer

162 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

**2**answers

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**

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**0**answers

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**

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**0**answers

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**

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**0**answers

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

**2**answers

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

**1**answer

552 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

**1**answer

460 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

**1**answer

382 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**

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**0**answers

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 ...

**7**

votes

**1**answer

289 views

### Isomorphic simple groups

It is known that $SL_{4}(\mathbb{F}_2)\cong A_8$. Obviously, this is equivalent to the existence of a subgroup of $Sl_4(\mathbb{F}_2)$ of index $8$. How to find such a subgroup?

**16**

votes

**1**answer

908 views

### Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order ...

**1**

vote

**2**answers

376 views

### finite abelian p-groups with solvable automorphism group

Let $G$ be an abelian (not elementary) finite $p$-group. In what conditions the automorphism group of $G$ is solvable?

**1**

vote

**1**answer

148 views

### Abelian p-subgroups of PSL(3,p^n)

In a note I saw this fact that $PSL(3,q)$ where $q=p^n$ does not have any abelian subgroup of order $q^3$. But I could not prove it or find any reference about it, could you please help me about it?
...

**2**

votes

**3**answers

263 views

### Finite / uniquely divisible abelian groups

Is there any counter example for the following statement?
STATEMENT:
Let $0 \to F \to A \to Q \to 0$ be a short exact sequence of abelian groups.
Assume that $F$ is a finite group, and $Q$ is a ...