# Tagged Questions

The finite-groups tag has no wiki summary.

**4**

votes

**1**answer

187 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 ...

**9**

votes

**0**answers

136 views

+100

### $2$-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, ...

**4**

votes

**2**answers

227 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 ...

**7**

votes

**0**answers

223 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, ...

**9**

votes

**0**answers

172 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 ...

**1**

vote

**1**answer

151 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 ...

**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 ...

**0**

votes

**0**answers

71 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 ...

**3**

votes

**1**answer

375 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), ...

**3**

votes

**0**answers

52 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) ...

**11**

votes

**1**answer

398 views

### Are There Always Group Generators Which Give Unimodal Growth?

Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function?
Background:
The counting function, $f(n)$, is a ...

**-1**

votes

**1**answer

83 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?

**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

**1**answer

65 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 ...

**0**

votes

**0**answers

100 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 $A=Gal(Q_n/Q)$. $A$ ...

**2**

votes

**0**answers

211 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 ...

**0**

votes

**0**answers

97 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 ...

**1**

vote

**0**answers

31 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 ...

**7**

votes

**2**answers

519 views

### What groups have a second maximal subgroup below exactly four maximal subgroups?

I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal ...

**10**

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 ...

**5**

votes

**1**answer

204 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 ...

**6**

votes

**1**answer

205 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 ...

**15**

votes

**1**answer

333 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 ...

**1**

vote

**1**answer

185 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**

votes

**2**answers

238 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?

**4**

votes

**2**answers

166 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 ...

**1**

vote

**0**answers

126 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 ...

**6**

votes

**1**answer

244 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 ...

**1**

vote

**1**answer

116 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)?

**2**

votes

**1**answer

201 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?

**5**

votes

**3**answers

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 $P$?
...

**11**

votes

**1**answer

457 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 ...

**2**

votes

**1**answer

318 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 ...

**4**

votes

**0**answers

68 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
...

**1**

vote

**1**answer

179 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 ...

**4**

votes

**0**answers

255 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 ...

**0**

votes

**0**answers

80 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

**7**

votes

**2**answers

343 views

### How many finite loops?

How many finite loops of order $n$ are there?
I am interested in the exact values of $n$ if $n <40$ or even reasonable estimates. I am also interested in formulae or bounds for all $n$.
Note ...

**2**

votes

**0**answers

190 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

148 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 $ ...

**5**

votes

**1**answer

335 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 $G$ ...

**3**

votes

**1**answer

180 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 ...

**3**

votes

**2**answers

266 views

### n-cocycles of finite abelian groups from cohomology group

Question: Given a generic finite abelian group $G=\mathbb{Z}_{N^{(1)}} \times \cdots \times \mathbb{Z}_{N^{(k)}}$.
(1) What is the explicit forms of its cohomology group (see my definition) in a ...

**2**

votes

**1**answer

206 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 ...

**11**

votes

**1**answer

233 views

### Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, g_{k+1}) ...

**3**

votes

**0**answers

217 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

710 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

**21**

votes

**0**answers

640 views

### Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,
$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...

**6**

votes

**5**answers

367 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 ...

**14**

votes

**2**answers

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 $G$ ...