The finite-groups tag has no wiki summary.

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**1**answer

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### Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...

**32**

votes

**3**answers

586 views

### Word evaluating to a group element and its inverse with different frequency

I'm supervising an undergraduate research project. Among other things, I've got the student to look at this paper of Gene Kopp and John Wiltshire-Gordon. This question arose from a missing complex ...

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**0**answers

233 views

### Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...

**19**

votes

**2**answers

696 views

### Definition of “finite group of Lie type”?

The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from ...

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**0**answers

89 views

### Soluble subgroups of 8-dimensional orthogonal groups over GF(4) transitive on nondegenerate 1-subspaces

Let $V$ be an $8$-dimensional vector space over $GF(4)$ equipped with a nondegenerate plus type quadratic form, $G$ be an almost simple group with socle $L=\Omega^+(V)$, and $H$ be a soluble subgroup ...

**2**

votes

**1**answer

130 views

### Subgroup structure of orthogonal groups of small dimension over finite fields

How much is known about the subgroup structure of the orthogonal groups (of dimension n<=7, say) over finite fields? Can anyone point me in the direction of a good reference? I'm aware of a book by ...

**4**

votes

**1**answer

87 views

### Generalized system normalizer for insoluble finite groups

For a finite group $G$ is there a subgroup $H$ such that for every chief factor $K/L$ of $G$ one has:
$G = K C_G(K/L)$ and $K \leq HL$ (so $K/L$ is inner and covered by $H$)
$G \neq K C_G(K/L)$ and ...

**2**

votes

**1**answer

193 views

### An upper bound for the maximal subgroups at fixed index?

Let us call a subgroup an injective homomorphism between groups.
I warn the reader that a subgroup designates here an inclusion $(H \subset G)$, not $H$ alone.
A subgroup $H \subset G$ is ...

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votes

**3**answers

1k views

### Is there a purely group-theoretic reformulation of an equivalence of subgroups?

There is an equivalence relation between inclusion of finite groups coming from the world of subfactors:
Definition: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset ...

**3**

votes

**1**answer

191 views

### non-split extension and Schur multiplier

Let $G$ be a central extension of the group $K$ by the group $H$. If we know that this extension is non-split, is it true that the order of $K$ must divide the Schur multiplier of the group $H$?

**4**

votes

**1**answer

602 views

### Finite groups with no elements of order $p^2q$

Let $G$ be a finite group. What can be said if $G$ has the following Property P: $G$ has no element of order $p^2q$ for any two distinct primes $p,q$?
In particular, which finite simple groups ...

**2**

votes

**1**answer

124 views

### Embeddings of of quotient singularities

Let $G$ be a finite subgroup of $SL(n,\mathbb{C})$. Let $G$ act on $\mathbb{C}^{n}$ with the action induced by $SL(n,\mathbb{C})$ and the induced action of $G$ on the unit sphere $S^{2n-1}$ is free. ...

**12**

votes

**1**answer

251 views

### Largest subset of $GL_n(p)$ in which pairwise subtraction is also in $GL_n(p)$

Suppose $X\subset \mathrm{GL}_n(p)$ is a set of invertible matrices such that for every $A,B\in X$ then also $A-B\in \mathrm{GL}_n(p)\cup \{0\}$. (If anyone knows a name for such sets I would be ...

**7**

votes

**1**answer

305 views

### Solvability of finite groups of order coprime to 15 — proof without using CFSG?

Is the solvability of finite groups of order coprime to 15
essentially easier to prove than the entire Classification of Finite Simple Groups?

**0**

votes

**1**answer

215 views

### For any n and some prime p there is an elemnet in Zp* of order n [closed]

How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...

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votes

**1**answer

81 views

### On Groups of Maximal Class: Reference

I will be happy if one gives references (oncluding current research) for `classification' (structure) of $p$-groups of maximal class which contain abelian maximal subgroup (i.e. abelian subgroup of ...

**4**

votes

**1**answer

306 views

### Does there exist an order in a number field of deg>1 with a map to F_p for all p?

This question is motivated by a computational issue. Suppose $R$ is a product of orders in numberfields such that there is no ring homomorphism $R \to \mathbb Z$, then can one write an algorithm that ...

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votes

**2**answers

200 views

### Principal series of finite group of Lie type

I have a naive question on complex representations of finite groups of Lie type.
Let $\bf G$ be a reductive group (say connected, with connected center, for safety)
defined over a finite field ...

**2**

votes

**3**answers

670 views

### General bound for the number of subgroups of a finite group

I am interested in the following:
Let $G$ be a finite group of order $n$. Is there an explicit function $f$ such that
$|s(G)| \leq f(n)$ for all $G$ and for all natural numbers $n$, where $s(G)$ ...

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votes

**1**answer

242 views

### Counting conjugacy classes in simple groups of Lie type

Finite groups of Lie type include those obtained as rational points of a connected simple algebrraic group over a finite field $k = \mathbb{F}_q$ of characteristic $p$: these are split or quasi-split. ...

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votes

**2**answers

166 views

### Conjugacy classes of the Ree groups

Dear all,
Consider the finite simple Ree groups $G=^2\hspace{-1mm}G_2(3^{2n+1})$ where $n$ is a positive integer. I would like to know the orders of conjugacy class representatives of $G$ and from ...

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votes

**1**answer

311 views

### Example of a finite group

Hi everyone,
I am looking for an example of finite group $G$ such that (a) the number of elements of order $2$ group $G$ is $p(p+1)/2$ or $p(p-1)/2$ ($p$ is a prime divisor of order $G$) (b) the ...

**2**

votes

**3**answers

400 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 $p$.

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votes

**2**answers

476 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

**1**answer

161 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 ...

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vote

**0**answers

146 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

**2**answers

316 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 ...

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votes

**2**answers

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 ...

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**2**answers

356 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 ...

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**0**answers

118 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 ($1/l+1/m+1/n<1$) ...

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votes

**1**answer

405 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 ...

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votes

**4**answers

565 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 ...

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votes

**2**answers

340 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 $W$?

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109 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

204 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?

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**0**answers

243 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?

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votes

**2**answers

351 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 $\pi(G)$ ...

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votes

**2**answers

556 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

336 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 ...

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votes

**1**answer

93 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 ...

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votes

**1**answer

249 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 ...

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**2**answers

425 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 ...

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**0**answers

151 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:
...

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votes

**3**answers

547 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?
...

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**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 ...

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votes

**1**answer

93 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
$$ \{e\}=G_0 \subsetneq G_1 \subsetneq \ldots \subsetneq G_{d-1} \subsetneq G_d=G. $$
...

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vote

**1**answer

547 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 ...

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vote

**1**answer

222 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 = ...

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votes

**2**answers

270 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 ...

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**0**answers

106 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 ...