Questions on group theory which concern finite groups.

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### What are the rank 3 boolean intervals [H,G], with G simple group?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:
Question: What are the rank $3$ boolean intervals of the form $[H,G]$, ...

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### a modular character problem [closed]

Let $B\in$Bl$(G|D)$ and suppose that $\sigma\in$Aut$(G)$ fixes every $\chi\in$Irr$(B)$. If $d\in D$, show that $d$ and $d^\sigma$ are $G$-conjugate. It is a problem from Navarro's book "characters and ...

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### Finite groups are isomorphic [closed]

For two finite groups $G_1, G_2$ if for every integer $n\geq 0$, $|G_1^n| = |G_2^n|$, then is it true that $G_1\cong G_2$? By $G^k$ we mean set $\{g^k|g\in G\}$.

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### When is the restriction map $res:H^2(G,U(1))\to H^2(Z_p\times Z_p,U(1))$ not the zero map?

Consider $G$ to be a finite group with non-trivial Schur Multipler $H^2(G,U(1))$, where $G$ acts trivially on the circle group $U(1)$.
By Example of a Schur-nontrivial group with no abelian subgroup ...

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244 views

### Simple groups and irreducible characters of degree 3

I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here.
The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}...

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**1**answer

160 views

### Primes and the factorial

Conjectures:
For all prime numbers $p>5$ there exist a prime number $q<p$ such that
$q\equiv m!\!\pmod p$, for some $m$ with $2<m<p$.
and
For all primes $p$ there is a prime $...

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### Group of Lie type as expanders: explicit estimates

In a paper Finite simple groups as expanders by M. Kassabov, A. Lubotzky and N. Nikolov there is a theorem, stating that there exists $\varepsilon>0$ and $k\in\mathbb{N}$, such that for every non-...

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130 views

### p-groups with special property on its centralizers

Thanks for any help or comment.
Suppose $G$ is a finite non-abelian p-group. Suppose $G$ has a proper non-abelian subgroup $M$ such that for every non-central element $x\in M$, $C_G(x)\subseteq M$. ...

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169 views

### representation of a group and its center

(I asked the following question at StackExchange but received no answer.)
Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional ...

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### Analogues of Goursat's Lemma for Infinite Products of Topological Groups

Recall that Goursat's Lemma has the following useful consequence. Let $G_1, G_2$ be finite groups with no common simple non-abelian quotients, and suppose $\gcd(|G_1^{\operatorname{ab}}|, |G_2^{\...

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154 views

### The number of fixed points of an automorphism of $\mathbb{Z}_m\times\mathbb{Z}_n$

Let $m$ and $n$ be two positive integers such that the groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have no common direct factor. Then an automorphism $f$ of $\mathbb{Z}_m\times\mathbb{Z}_n$ is of type
$$\...

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**1**answer

103 views

### Bounding the union of conjugates of a maximal subgroup of the Symplectic group over a finite field

Let $g \geq 1$ be a positive integer, and let $p$ be a prime. Consider the symplectic group $G := \operatorname{Sp}_{2g}(\mathbb{F}_p)$ of symplectic matrices with entries in $\mathbb{F}_p$. Let $M \...

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133 views

### Ring structure on cohomology of groups

Assume that $G$ is a finite group and that $A$ is an arbitrary $G$-module. Then we know that can define the cohomology groups of $G$ with coefficients in $A$ in the usual way and we denote the latter ...

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212 views

### Exceptional isomorphisms between finite simple Chevalley groups

Steinberg's "Lectures on Chevalley Groups"
https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf
contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...

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**1**answer

154 views

### Sum of Young symmetrisers of a given shape

Preliminaries and notation:
Let $n\in \mathbb{Z}_{>0}$ and $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_s)\vdash n$ be a partition. Given a Young diagram of shape $\lambda$, we can associate it ...

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236 views

### Is $(G\rtimes H,H)$ a Gelfand pair iff $G$ is abelian?

Let $G$ be a finite group and $K \subset G$ a subgroup. Then $(G,K)$ is a Gelfand pair if the double coset Hecke algebra $\mathbb{C}(K \backslash G / K)$ is commutative.
Let $H$ be a subgroup of $...

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143 views

### Do the irreducible modules of this finite group preserve a tensor product structure?

I am interested in a particular group $G$, where
$$ (A_4\times C_\ell) \lhd G \lhd S_4 \times D_\ell$$
Here, $C_\ell$ is cyclic, $D_\ell$ is dihedral of order $2\ell$, and the two inclusions both have ...

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110 views

### On self-dual group-subgroup subfactors

Let $(N \subset M)$ be a finite index inclusion of ${\rm II}_1$ factors, and $N \subset M \subset M_1$ the basic construction.
The subfactor $(N \subset M)$ is called self-dual if it is isomorphic to ...

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462 views

### Non-vanishing of the Tate-Shafarevich kernel in group cohomology

Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$).
We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional ...

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170 views

### Subgroup ranks of the symmetric group

It's well known that every subgroup $G$ of $S_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter)
I have heard many times a ...

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### Is there an interval of finite groups, at index n, with strictly more elements than the subgroup lattice of any group, of order n?

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice.
Let $s(n):= max\{|\mathcal{L}(G)| \text{ for } |G|=n \}$.
There is an OEIS page for the sequence $s(n)$: A018216
1, 2, 2, 5, 2, ...

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### Can a finite group have its third cohomology determined by a quotient group?

Does there exist a finite group $G$ satisfying either of the following?
$G$ admits a non-trivial, proper, central subgroup $A$ such that every (normalized) 3-cocycle of $G$ is cohomologous to a (...

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242 views

### Is it known whether every symmetric pair of finite groups of Lie type is a Gelfand pair?

A pair of groups $(G,H)$ is called a symmetric pair if $H$ is the group of fixed points of an involutive automorphism of $G$, for example $(GL(2n,\mathbb{F}_q),Sp(2n,\mathbb{F_q}))$ is a symmetric ...

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230 views

### A good upper-bound for the cardinal of an interval of finite groups

This post is a relative version of General bound for the number of subgroups of a finite group
Let $[H,G]$ be a interval of finite groups with $|G:H| = n$.
Question: What is a good upper-bound of $|[...

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### Is there a nice explanation for this curious fact about cyclic subgroups?

Here's something that I noticed that quite surprised me.
Let $G$ be a finite abelian group. Consider the following expression.
$$
\nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H|
$$
It ...

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### Nonvanishing of the dual Euler totient on boolean intervals of finite groups

The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$.
Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...

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### Primary invariants

This question is related to the earlier question which is in the given link:
Primary invariants of a finite group
Let $G$ be a finite group and $V$ a complex representation of degree $n$, and let $...

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### Chief factors and local formation

Every thing below is concerned with finite groups.
My question is about this paper
A class of groups is a collection $\mathcal{X}$ of groups with the property that
if $G \in \mathcal{X}$ and if $H \...

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### Finite groups inside an infinite group with the same homology

Suppose we have a triple of groups $G,H,K$ verifying the follwing conditions
$G$ and $H$ are finite groups and $K$ an infinite group.
there exists two monomorphisms $G\rightarrow K\leftarrow H$ ...

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### Computing characters of $\alpha$-projective representations

Given a finite group $G$, a finite cyclic group $A$ (viewed as a subgroup of $\mathbb{C}^{\times}$, i.e. generated by a $|A|$-th root of unity), and a 2-cocycle $\alpha\in Z^{2}(G,A)$. Recall that an $...

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### Primary invariants of a finite group

For a finite group $G$ and complex representation V of degree $n$, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants ...

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### Quotient of Coxeter complex in terms of double cosets?

In Victor Reiner's Quotients of Coxeter Complexes and $P$-Partitions, we have the below definition for the quotient complex of a Coxeter complex by a finite subgroup of the Coxeter group. I think ...

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### Finite groups with lots of conjugacy classes, but only small abelian normal subgroups?

Denote the commuting probability (the probability that two randomly chosen elements commute) of a finite group $G$ by $\operatorname{cp}(G)$. By a result of Gustafson [2], $\operatorname{cp}(G)=\...

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### Known results in the Cohomology of finite groups

I am learning to compute cohomology of finite groups and came across this survey article http://www.ams.org/notices/199707/adem.pdf "Recent Developments in the cohomology of finite groups" by ...

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### For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]:
If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free $...

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163 views

### Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...

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### Radical and Centric not Essential P-group

I'm looking that in the Fusion System categories, the p-subgroups that are essential, are centric (by definition) and radical (by implication of the definition), but I want to know if there is an ...

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326 views

### Character Values for Alternating Groups of degree $\geq 7$

Is it true that in each row and column of the character table of alternating groups with degree $\geq 7$ there are at most two complex values? Any reference will be highly appreciated.

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### Group cohomology of the cyclic group

It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution,
$\require{AMScd}$
\begin{CD}
\cdots @>N>> \mathbb ...

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484 views

### A question about (unicity of certain cycles in a Cayley graph of a) symmetric group

Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus $$[(1,2,\ldots,n)(1,2)]^{n-...

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### p-groups as finite union of disjoint normal abelian subgroups

I was interested in knowing if groups with following property have been studied( like what can be said about structure of the group) :
"$G$ can be written as disjoint union of a given number of ...

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69 views

### Exterior Powers of finite abelian group

Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map
\begin{align}
\alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\
a_1\...

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230 views

### Finite group action on quasi-projective varieties

Let $X$ be a smooth, quasi-projective variety, $G$ be a finite group which acts freely and properly on $X$. Denote by $\alpha:X \to X/G$ the quotient. Is $\alpha$ generically etale?
Also, as I am ...

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170 views

### Is there a characterization of CI-groups of order less than 100?

We know some benefit criterion in articles such as:
C. H. Li, On isomorphisms of finite Cayley graphs-a survey, Discrete Math., 256 (2002) 301-334.
C. H. Li, Z. P. Lu, P. P....

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### Is there a formula for the number of elements in $S_n$ having length $k$ with respect to the generators taken to be the transpositions?

Define the length of a permutation as the minimum number of adjacent transpositions needed to describe it.
We know that there is only one element of length $0$ in $S_n$ and $n-1$ elements of length $1$...

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### A finite supersolvable group with generators of prescribed order

Let $G=\langle a,b\rangle$ be a finite supersolvable group. Is there any special information about the structure of $G$ when $o(a)=2$ and $ o(b)=2^k > 2$?

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476 views

### Transitive permutation groups which all of their proper subgroups are intransitive

Let $G$ be a transitive permutation group on a finite set $\Omega$. It is clear
that if $G$ is regular, then every proper subgroup of $G$ is intransitive. Is
there any other class of groups with this ...

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40 views

### Gramian of a permutation group orbit

let $W\in R^{d\times k},d>k$. Suppose that the associated gramian has the following structure:
$$
W^TW=(P_{1}t,\cdots,P_{k}t)
$$
with the set $\{P_{i},i=1,\cdots,k\}$ forming a group of ...

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### An optimal lower bound related to generators in a boolean interval of finite groups

Let $[H,G]$ be a rank $n$ boolean interval of finite groups (i.e. $[H,G] \simeq B_n$ as lattice).
Let the set $E = \{ g \in G \ | \ \langle H,g \rangle = G \}$
Remark: If $g \in E$ then $Hg \...

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132 views

### Lifting automorphisms of quotient groups

I am concerning here a natural question:
Problem: Let $G$ be a finite group, and let $N$ be a characteristic subgroup of $G$. When can an automorphism $\varphi\in\mathrm{Aut}(G/N)$ be lifted to an ...