The finite-groups tag has no wiki summary.

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### Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...

**4**

votes

**1**answer

166 views

### Maximum length of chains of subgroup in GL(n,q)

Let G=GL(n,q) be a general linear group n-dimensional over a field with q element (q power of a prime). I am looking for an estimate of maximum length of chains of subgroup in G.
Thanks.

**2**

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**0**answers

86 views

### Coprime automorphisms of finitely generated pro-$p$ groups

Let $P$ be a finitely generated pro-$p$ group and let $G$ be a semidirect product $P \rtimes A$, where $A$ is a finite group of order coprime to $p$ that acts faithfully on $P$. Then one can show ...

**2**

votes

**1**answer

326 views

### Groups like symmetric group

For sufficiently large n consider this question
Let $G$ be a finite group with the following properties:
$|G|=n!$
$H,K$ are subgroups of $G$ such that $H\cap K=1$ and $H\cong S_{3}$ and $K\cong ...

**3**

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**0**answers

94 views

### Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?

**2**

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**1**answer

122 views

### generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?

**4**

votes

**1**answer

277 views

### Which finite groups can be characterized by their subgroup orders?

Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? ...

**6**

votes

**2**answers

266 views

### Subgroups from which all class functions extend to class functions on the ambient group

Until someone suggests better terminology, let me call a subgroup H of a finite group G segregated if every class function on H can be extended to a class function on G. Equivalently, H should have ...

**23**

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**1**answer

624 views

### Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of ...

**0**

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**0**answers

66 views

### What algebras does the hidden subgroup problem for finite abelian groups apply to?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ...

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**2**answers

243 views

### Bound for the Frattini subgroup of a $p$-group

Assume that $G$ is a finite $p$-group, $p$ odd, with a non-trivial elementary abelian Frattini subgroup. Then both $\Phi(G)$ and $G/ \Phi(G)$ are vector spaces over $\mathbb{F}_p$. Is it possible to ...

**4**

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**1**answer

257 views

### Normal Subgroup Growth

Let $F$ be a free group on $d$ generators.
Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$?
Explicitly, for each natural number ...

**4**

votes

**2**answers

247 views

### Simple groups and words

Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting ...

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279 views

### $2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...

**9**

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**0**answers

237 views

### Non split extension isomorphic (as a group) to a split extension

$\def\Z{\mathbb{Z}}$
Let $A$ be a finite abelian group and $G$ a finite group acting on $A$.
Then the extension $0 \to A \to E \to G \to 1$ is splits if and only if the corresponding $2$-cocycle is ...

**7**

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251 views

### Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...

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**0**answers

108 views

### Two $p$-groups whose automorphism groups have isomorphic Sylow $p$-subgroups

Fix a prime $p$, and let $M$ be the unique nonabelian group of order $p^3$ and exponent $p$. Let us denote by $E_n$ the elementary abelian group of rank $n$.
Is it true that $\operatorname{Aut}(M ...

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votes

**1**answer

101 views

### Action of automorphism group on Lie algebra [closed]

I want to know whether an automorphism group of a simple Lie algebra over $GF(2)$, acts transitively on non-zero elements of Lie algebra or not? How can I check this property?

**1**

vote

**1**answer

74 views

### Control of $p$-extensions by subgroups of index coprime to $p$

Let $G$ be a finite group and let $M$ be a $G$-module that is a finite abelian $p$-group. Suppose we have extensions
$1 \rightarrow M \rightarrow E_1 \rightarrow G \rightarrow 1$
and
$1 ...

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**0**answers

106 views

### a problem in character theory of finite group

I want to ask a question on character theory of finite group.
Let $G$ be a finite nonablian group， and let $p$ be the minimal divisior of |G|=n. Suppose $\chi\in IRR(G)$. Let $A=Gal(Q_n/Q)$. $A$ ...

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105 views

### Orbits of stabilizer of two points in a 2-transitive permutation group

I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their ...

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37 views

### Finite quasigroup not coming from a scrambled finite group

Given any finite group with multiplication $m(-,-)$ and three permutations $p,q,r$ on the underlying set of the group, we can obtain a quasigroup with binary operation $g*h:= p(m(q(g),r(h)))$. What is ...

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**1**answer

259 views

### In which fixed-point free representations is the sum of every 3 elements invertible?

A representation $\rho:G\to GL_k(\mathbb{F})$ is called fixed-point free if for every $1\neq g\in G$ and every $0\neq v\in \mathbb{F}^k$, $\rho(g)v\neq v$. Stated differently, it is a representation ...

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**1**answer

234 views

### Primitive, non-2-transitive groups with very large orbitals?

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not ...

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vote

**1**answer

198 views

### On the Complement of a subgroup

This question was asked in
http://math.stackexchange.com/questions/729648. Since I did not get any answer I am asking it here.
In an answer in Mathoverflow I see an answer but I could not ...

**4**

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**2**answers

259 views

### Real representation of group of odd order

Let $G$ be a finite group of odd order. Suppose that $G$ has a real 4-dimensional faithful representation. Is it true that $G$ should be abelian in this case?

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**1**answer

403 views

### Finite groups $G$ so that $G$ has exactly two subgroups of a given order

Is there a finite group $G$ and a divisor $d$ of $|G|$ so that $G$ contains exactly two subgroups of order $d$?
The motivation for this question is an old qual problem (see ...

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176 views

### Are the following Cayley digraphs Hamiltonian?

Consider the Cayley graphs $A'G_n$ on the alternating group $A_n$ with generating set $S = \{(1i2) : 3 \leq i \leq n\}$, for $n \geq 4$.
See the following page on Alternating Group Graphs for ...

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vote

**1**answer

191 views

### Existence of homogeneous single chain compositions of a given maximal subfactor?

All the subfactors here are irreducible inclusion of hyperfinite II$_1$ factors.
A subfactor $(N \subset M)$ is Homogeneous Single Chain ($HSC$) if its lattice of intermediate subfactors is a ...

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**0**answers

82 views

### Double coset relation for unique intermediate subgroup (with homogeneity)

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$) and $(H \subset K) \sim (K \subset G)$ (homogeneity) ...

**3**

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**1**answer

437 views

### On a metric associated to certain finite groups

Let $G$ be a finite group. Define $d:G\times G\longrightarrow\mathbb{N}$ by $d(x,y)=o(xy^{-1})-1, \forall\, x,y\in G$. Then $d$ is a metric on $G$ if and only if
$$(*)\hspace{5mm}o(ab)<o(a)+o(b), ...

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vote

**1**answer

123 views

### Max order for which connected Cayley Graphs are known to be Hamiltonian

There is a well-known conjecture that all connected Cayley graphs are Hamiltonian.
For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)?

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238 views

### Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there.
Recall that a subfactor is Dedekind if all its intermediate subfactors are normal.
A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...

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**1**answer

488 views

### Find finite groups $G\cong Aut(G)$

I become interested in this problem because $G\cong Aut(G)$ suggests a special symmetry in $G$.
From $G/Z(G)\cong Inn(G)$ we know complete group is the anewer for the simplest case, though this class ...

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78 views

### Automorphisms with an orbit “transversal” to a subgroup

Let $G$ be a finite group, $H$ be a subgroup of index $2$, $\phi\colon G\to G$ an automorphism. For $d\in \mathbb N$ let us say that $\phi$ is $d$-transversal to $H$ iff there is $h\in H$ such that
...

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**1**answer

194 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 ...

**0**

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**0**answers

87 views

### Brauer characters of finite simple group $E_8(5)$

I would like to find the irreducible characters of the group $E_8(5)$ (mod 2)?
Can anyone help? (I am elementary in working with Brauer characters)
Many thanks

**1**

vote

**1**answer

177 views

### List of Conjugacy Classes of the General Linear group over $\mathbb{Z}/p^2\mathbb{Z}$

Does anybody know the answer to this or a good way to go about working this out? I have a list for $GL_2(Z/pZ)$ and I am trying to lift it to this; I have mostly been using fairly elementary algebraic ...

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**1**answer

187 views

### on the solvable groups of order $p^aq^b$ [closed]

We know that if $ p$ is a prime number then $ O^p (G) $ is the smallest normal subgroup of $ G $ such that $ G/O^p (G) $ is a $ p $-group.
Now let $ G $ be a finite group of order $ p^aq^b $ where $ ...

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**1**answer

210 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?

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196 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 ...

**5**

votes

**1**answer

356 views

### Finite groups for which the element orders form an arithmetic progression

Which are the finite groups $G$ such that the element orders of $G$ form an arithmetic progression? Several remarks:
$S_3$, $A_4$ and any $p$-group of exponent $p$ satisfy this property.
If $G$ ...

**6**

votes

**1**answer

267 views

### The best upper bound for the number of involutions in a finite non-abelian simple group

Let $G$ be a finite non-abelian simple group and $t$ is equal to the number of involutions of $G$. We know that $t<|G|/3$ or $3t+1 \leq |G|$. Is this the best upper bound for the number of ...

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315 views

### More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms

This question is a follow-up to
Monstrous Moonshine for Thompson group $Th$?
and is based on various comments to that question, in particular S. Carnahan's mention of the
connection to known ...

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**1**answer

340 views

### A second isomorphism theorem for the inclusions of groups

The usual second isomorphism theorem for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq ...

**2**

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211 views

### on the prime divisors of $(p^2+1)/2 $

The following question is equivalent to a problem in group theory.
Let $ p > 13$ be a prime number distinct from 239. Let $ a=(p^2+1)/2 $. Is there any prime divisor $r$ of $a$ such that $r\mid ...

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votes

**1**answer

204 views

### Normal intermediate subgroup and normal core

Let $G$ be a finite group and $H$ a subgroup.
The normal core of $H$ in $G$ is $core_G(H) := \bigcap_{g \in G}g^{-1}Hg$
Definition: $K$ is a normal intermediate subgroup of the inclusion $(H ...

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**0**answers

234 views

### On the Groups of Order $(p^2+1)/2$

A few days ago I asked a question (Groups of order $p(p^2+1)/2$) about a finite group of order $p(p^2+1)/2$ and I got a lot of useful information about it. Thanks for the nice and very helpful ...

**5**

votes

**5**answers

900 views

### Groups of order $p(p^2+1)/2$

It seems that when $p>3$ is a prime, then each group of order $p(p^2+1)/2$ is abelian as I checked by Gap for small $p$. Is it true for each $p$?
Thanks for your answers

**6**

votes

**5**answers

443 views

### Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...