Questions on group theory which concern finite groups.

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### Normal conjugate of elements of unipotent upper tringular matrices over F_q

Let $UT_n(q)$ be the group of upper triangular matrices with entries in the finite field $F_q$ and ones on the diagonal. Denote the normal closure of an element $s\in UT_n(q)$ by $s^{UT_n(q)}$, i.e., $...

**6**

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**4**answers

661 views

### When does $\langle grg^{-1}|g\in G\rangle = \langle gsg^{-1}|g\in G\rangle$ imply $\langle r\rangle=\langle xsx^{-1}\rangle$?

We call a finite group $G$ normal if for all $s,r\in G$, if $\langle gsg^{-1}:g\in G\rangle =\langle grg^{-1}:g\in G\rangle $ then there exists $x\in G$ such that $\langle r\rangle =\langle xsx^{-1}\...

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**1**answer

121 views

### Are rotational isometries groups generated by some kind of rotations?

If we consider the index $2$ subgroup of a Weyl group consisting of the isometries with determinant $1$ (the 'special' Weyl group), is it known that it is generated by rotations around some fixed axes?...

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**1**answer

60 views

### Finite groups of planar homeomorphsims

Let G be a finite subgroup of the group H of orientation-preserving homeomorphisms of the plane that fix the origin. Is G conjugate in H to a group of rotations?
I've been told this result was ...

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**1**answer

51 views

### How to claculate the $T$-stable subgroup of second cohomology group

Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1.
Let $N=\langle y,w \rangle \...

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**1**answer

220 views

### Finite groups of order $n$ having exactly $n$ subgroups

Is it known a characterization of finite groups of order $n$ having exactly $n$ subgroups?
A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}_2$ with ...

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**1**answer

106 views

### Generalization of a lemma of Livne

Let $H$ be a finite $2$-group. Let $N_{4}(H)$ be the subgroup generated by fourth powers. Let $H_{4}$ be the last term in the short exact sequence $1\rightarrow N_4(H) \rightarrow H \rightarrow H_{4} \...

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**3**answers

413 views

### Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$

It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$
It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...

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**1**answer

122 views

### Generate harmonic polynomials for a finite group

Let $G$ be a finite group acting on a complex vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A ...

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**1**answer

141 views

### The groups with nilpotent Hall $p'$ subgroup

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elements.
Theorem $2$: A group of order $p^nq^m$ is solvable.
Theorem $1$ depends on ...

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**0**answers

156 views

### Are all sneaky groups products of Frobenius and 2-Frobenius groups?

I've been stuck thinking about this for a while.
Def. Let $G$ be a finite solvable group whose order is divisible by only three primes: $p,q,$ and $r$. Suppose that $G$ has cyclic subgroups of ...

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**1**answer

1k views

### Are there $n$ groups of order $n$ for some $n>1$?

Given a positive integer $n$, let $N(n)$ denote the number of groups of order $n$, up to isomorphism.
Question: Does $N(n)=n$ hold for some $n>1$?
I checked the OEIS-sequence https://oeis.org/...

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**1**answer

175 views

### Strongly real elements of odd order in sporadic finite simple groups

Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.
Question: Is it true ...

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**1**answer

459 views

### Okounkov-Vershik approach to representation theory of $S_n$

This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S_n$ is all about. It's ...

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**1**answer

251 views

### number of maximal subgroups of the symmetric group

What is the asymptotics of the number of the maximal subgroups of $S_n$ (as a function of $n$)? This must be written down somewhere...
EDIT I am actually more interested in the number of conjugacy ...

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**3**answers

500 views

### The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$.
I am interested in the following questions:
How ...

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**1**answer

136 views

### Determining conjugacy class of a subgroup from the sizes of its intersections with the conjugacy classes

Let $C_1,C_2,\ldots,C_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C_i|=|K \cap C_i|$ for ...

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96 views

### Expository papers for Feit–Thompson Theorem [duplicate]

Feit–Thompson theorem states that every finite group of odd order is solvable.
Its proof is near 300 pages so it is definitely not an easy paper to read without a prior knowledge about the general ...

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211 views

### An abstract zero-sum problem

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...

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113 views

### Centerless finite groups with exactly three prime divisors

Let $G$ be a centerless finitee group and $\pi(G)=\{p,q,r\}$ be the set of prime divisors of $|G|$. I have some questions about these groups.
Is there any classification of such groups?
In what ...

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**1**answer

126 views

### Is the Frattini subgroup of a free profinite group trivial?

Let $\widehat{F_2}$ be the free profinite group of rank 2. Is its Frattini subgroup $\Phi(\widehat{F_2})$ trivial?
I know that the Frattini subgroup of a pro-$p$ group is open. On the other hand, the ...

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229 views

### Representations of orthogonal groups over the field of two elements

I am looking for some references on modular representation theory of the orthogonal groups $O_{2n+1}(2)$, $O_{2n}^{+}(2)$, or $O_{2n}^{-}(2)$ over $\mathbb{F}_2$.

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440 views

### Minimal maximal subgroup of the symmetric group

The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we ...

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79 views

### Rank gradient in free products amalgamating a finite subgroup

Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$.
Must $A *_C B$ have positive rank gradient?
See Which 3-manifolds have ...

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359 views

### Characterization of finite groups using sum of the orders of their elements

Notation: If $G$ is a finite group, $o(g)$ denotes the order of the element $g\in G$.
Motivation: Some finite groups could be uniquely determined by the size of the group. For example given a prime ...

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186 views

### a question about a group which has a factor isomorphic to a permutation group

Thanks for any help or comments.
Suppose $G$ is a finite group and $t$ is an involution in $G$. Suppose also that
$\frac{C_G(t)}{\langle t\rangle}\cong \mathbb{S}_{n-2}$ and $C_G(t)$ contains some ...

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95 views

### When can the restriction map be zero or onto?

Let $G$ be a finite $p$ group of order $p^n$ and $N$ be a subgroup of $G$ of index $p$. Let $res:H^2(G,A) \rightarrow H^2(N,A)$ be the restriction map on cohomology, where $A$ is a trivial $G$ module. ...

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**1**answer

104 views

### Finding a semigroup that maximizes the trace of a sum of matrices

Let $H$ be a finite semigroup containing $n$ elements from a compact group $G$. I am trying to solve
$$\max_{h_i,\ h_j\ \in\ H} \operatorname{tr} \sum_{i,\ j\ \leq\ n} \rho(h_i^{-1} h_j)(A_j A_i^T)$$...

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70 views

### Salvaging Howson's theorem for free profinite groups

This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem?
Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...

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318 views

### Why does iterated indexing avoid cycles of length 5?

Start with a permutation $s_0$ of the numbers
$(1,\ldots,n)$, e.g., for $n=10$,
$s_0=(8,2,1,6,9,7,10,5,4,3)$.
Form $s_1$ by using the numbers in $s_0$ as indices into $s_0$.
So $s_1$ is composed of ...

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**1**answer

173 views

### Existence of some character degree in a solvable group

Let $p=2^n-1$ be a Mersenne prime. We know that $H=PSL(2,p^4)$ has an irreducible character of degree $ p^4$.
Is there any solvable group of order $H$ with an irreducible character of degree $ p^4$. ...

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80 views

### recovering information about a group from the spectrum of its Cayley graph

Suppose you have a finite group and you consider its Cayley graph with respect to some fixed generating set of nonidentity elements closed under inversion. Are there any results known to the effect ...

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291 views

### Faithful projective representations of symmetric groups

This is a reference request.
Do you know where I can find the dimensions of the faithful projective representations of $S_n$ and $A_n$ for $n\ge 5$?
Thank you in advance.

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557 views

### Beyond Brauer's theorem

Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary ...

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**1**answer

645 views

### “Squeezing” a finite group between symmetric groups

For a finite group $G$, take $m$ as the largest integer such that $G$ has a subgroup $H\cong S_m$ and $n$ as the smallest integer such that $G$ is itself isomorphic to a subgroup of $S_n$. We then ...

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473 views

### Counting cyclic subgroups of order $p^{2}$: $p$ an odd prime vs. $p=2$

While playing with Frobenius' problem (about finite groups $G$ in which, for some positive integer $n \mid |G|$, there are exactly $n$ elements of order dividing $n$), I came up with the following ...

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**1**answer

167 views

### Intersection of maximal subgroups of PSL(2,q)

Let $G := PSL(2,2^r)$, and let $M$ be a maximal subgroup of $G$ isomorphic to $PSL(2,2^s)$. I need to compute $H := M \cap M^g$ for $g \in G-M$. It seems to me that $|H|$ must be $2^r, 2^r\pm 1$ or $...

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943 views

### Number of squares in a finite group

This was asked at MSE but never answered.
Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if
...

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338 views

### The number of subgroups of ${\frak S}_n$

Because of my interest in this question, I listed the subgroups of ${\frak S}_n$ for $1\le n\le4$. I found that the number of subgroups are, respectively, $1,2,6,24$. It might be a coincidence, or it ...

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**1**answer

158 views

### Reference sought regarding the fact that all non-abelian finite simple groups are 2-generated

I've learned that all non-abelian finite simple groups are $2$-generated,
i.e. have a generating set of cardinality $2$.
Is there a reference to this statement which does not just point to the
...

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**1**answer

108 views

### positions of regular cubes in Euclidean space with all its vertices without distinction

Let $P$ be the standard regular cube, centered at the origin of $\mathbb{R}^3$ with all its vertices on the unit sphere.
If the vertices are labelled by $1,2,3,4,5,6,7,8$, then the collection of ...

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147 views

### Which Dihedral Groups are $\text{CI}$-Groups?

Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard.
Let $G$ be a finite group. A subset $S$ of group $G$ ...

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**1**answer

184 views

### Structure of Deligne-Lusztig representations $R_{T,\theta}$ for ministropic $T$ and cuspidal representations

Let $G$ be a reductive group over a finite field $k$, let $F$ be a Frobenius morphism on $G$.
I'll start with a somewhat vague question and make my question more specific further down:
How do ...

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75 views

### maximal elementary abelian p-subgroups of finite groups of Lie type

Let $G$ be a simple algebraic group over an algebraically closed field k of characteristic $p$. Let $G(\mathbb{F}_{p^r})$ be the corresponding finite group of Lie type. Is it true that every maximal ...

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69 views

### associativity of the extension of finie groups [closed]

Following my previous question I have two questions:
1.the extensions of the group is associative or not, i.e. as we know by the notations of atlas if $S={\rm PSL}(2,q)$, where $q=p^f>9$, then $2....

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**1**answer

150 views

### Extensions of $\Bbb Z_3$ by $PGL(2,q)$ where $q$ is odd

Let $q$ be odd. If $G$ is a finite group such that $G$ has a normal subgroup $H$ of order $3$ such that $G/H\cong {\rm PGL}(2,q)$, what can we say about $G$. Is it true in general that $G\cong {\Bbb Z}...

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287 views

### mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$.
(1). What is the cohomology ring
$$
H^*(A_4;\mathbb{Z}/3)
$$
and its Steenrod operation $P^i$'s?
(2). Are there general results about the ...

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109 views

### Exotic 2-adic lifts of mod $2$ Steinberg idempotent

Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...

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**1**answer

218 views

### To calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$. I wish to calculate $Tor_1^G(\mathbb{Z},N_{ab})$ and $Tor_1^Q(\mathbb{Z},N_{ab})$, where $Q=G/N$. I could not found any lecture notes ...

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**1**answer

79 views

### The maximal possible rank of a subgroup of a product of special linear groups

In this question I ask for a generalization of What is the maximal possible rank of a subgroup of a special linear group mod a prime?
Let $p_1, \dots, p_r$ be $r$ distinct odd primes.
Set $$G = \...