Questions on group theory which concern finite groups.

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799 views

### Can we bound degrees of complex irreps in terms of the average conjugacy class size?

This question arises when looking at a certain constant associated to (a certain Banach algebra built out of) a given compact group, and specializing to the case of finite groups, in order to try and ...

**11**

votes

**2**answers

878 views

### The maximum order of finite subgroups in $GL(n,Q)$

For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders.
It appear in many articles a ...

**11**

votes

**1**answer

414 views

### Group theory conjecture on hurwitz groups

Conjecture: Let $p$ be a prime.
Then the group
$G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^9, (([a,b]^4)b)^{2p} \rangle$
has a composition series of the form
${\rm PSL}(2,8) - {\rm Z}_p - ...

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votes

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1k views

### Number of generators of a subgroup of a finite simple group

For a finite group $G$ we denote $d(G)$ the minimal size of a set of generators of $G$. We define $D(G) = \max( d(H) \mid H\leq G)$.
Let $S$ be a finite simple group. Are there `good' bounds on ...

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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 ...

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votes

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760 views

### Iterated semi-direct products

Let $G$ be a finite group. Suppose that we can write $G= A \rtimes B$ and also $A = C \rtimes D$. Further suppose that C is normal in $G$ (not just in $A$). Then can we write $G = C \rtimes E$ where ...

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votes

**1**answer

413 views

### Is every finite group a proper quotient of a finite primitive group?

Let $G$ be a finite group. Is there necessarily a finite primitive permutation group $P$ and a normal subgroup $N>1$ of $P$ such that $P/N \cong G$?
If not, what restrictions are there on ...

**11**

votes

**1**answer

1k views

### Extension of induced reps over Z: is it a sum of induced reps?

Let $G$ be a finite group. If $L$ is a finite free $\mathbf{Z}$-module with an action of $G$, say $L$ is induced if it's isomorphic as a $G$-module to $Ind_H^G(\mathbf{Z})$ with $H$ a subgroup of $G$ ...

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votes

**1**answer

319 views

### Can a large transitive permutation group need many generators?

let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have ...

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**1**answer

488 views

### On the order of finite simple groups

About the order of finite simple groups there exists a very interesting result which stated as follows:
Let $G$ be an non-solvable simple group of order $g$. If $p\mid g$, where $p>g^{1\over 3}$ ...

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610 views

### Convenient reference for subgroups of a finite semidirect product?

Given a finite group $G= H \ltimes N$ (with no particular constraints on $H, N$), it's probably been known for a long time how to describe efficiently the possible subgroups of $G$. A graduate ...

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**1**answer

273 views

### A sum over characters of the symmetric group

Let $C_\mu$ be the size of the conjugacy class in $S_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S_n$.
Let ...

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votes

**1**answer

340 views

### A subgroup intersects conjugacy class of every prime power order element

Let $G$ be a finite group and $H$ be a subgroup of $G$. Suppose that for any prime power order element $x$ of $G$, there exists some element $g$ in $G$ such that $x^g$ is contained in $H$. Does it ...

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619 views

### Regular elements in the torus of a group of Lie type

Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...

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851 views

### “Antipodal” maps on regular graphs?

This question is related to Realizing the diameter of a finite regular graph
Let $X=(V,E)$ be a finite, connected, regular graph of diameter $D$. Assume that, for every vertex $x\in V$, there exists ...

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**1**answer

351 views

### What are smallest finite images of triangle groups?

Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, ...

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1k views

### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and ...

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votes

**1**answer

738 views

### Non-isomorphic two-transitive permutation groups with isomorphic point stabilizers

The permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}_8$ ...

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238 views

### The Weyl group of E8 versus $O_8^+(2)$

Right now Wikipedia says:
The Weyl group of $\mathrm{E}_8$ is of order 696729600, and can be described as $\mathrm{O}^+_8(2)$.
The second part feels wrong to me. $\mathrm{O}^+_8(2)$ is the ...

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votes

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1k views

### Mystery of the Monstrous Moonshine

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it,
...

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**1**answer

482 views

### complexity of counting homomorphisms

This is a question I have thought about and asked a number of people, but have never got an answer beyond "It should be true that..."
Given a finitely generated group $G$ (eg. a link group ...

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votes

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297 views

### Elementary consequences of commuting limits and colimits over groups

This is a crosspost of this MSE question.
In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...

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315 views

### Elements of order 3 normalizing no non-identity 2-subgroups in Almost Simple Groups

This question is partly motivated by a situation which arises in modular representation theory. A finite group $G$ is said to be almost simple if $G$ has a unique minimal normal subgroup which is a ...

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908 views

### Number of isomorphism types of finite groups

Are there some good asymptotic estimations for the number $F(n)$ of non-isomorphic finite groups of size smaller than $n$?

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356 views

### The cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$

Let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible ...

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297 views

### existence of a finite group which is the union of self normalizing subgroups

Can a finite group G be the union of self normalizing subgroups such that the intersection between any two of these subgroups is equal to the unit of the group G? I don't think so but I can't prove ...

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633 views

### A question on the set of element orders of a finite group

Let $G$ be a finite group of order $n$ and denote by $\pi_e(G)$ the set of element orders of $G$. What can be said about $G$ if $\pi_e(G)$ forms a sublattice of the lattice of divisors of $n$?

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608 views

### Automorphism Group of a p-group : Looking for a Reference

In the following post by DavidLHarden :
See Here
He quoted the following claim:
"There is a theorem that says that if $p$ is a prime and $|G|=p^n $ , then $|AutG| $ divides
$ \Pi_{k=0}^{n-1} ...

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688 views

### Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety?

Deligne-Lusztig theory
is awesome. You take a maximal torus $T$, you take a character $\theta$, construct a variety $X_T$$^*$, take etale cohomology, get a virtual character $R_T^\theta$, maybe it's ...

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votes

**1**answer

504 views

### Does every automorphism of G come from an inner automorphism of S_G?

I feel sort of silly asking this question. Unless I'm very much mistaken the paper I'm reading assumes the following statement:
Let $G$ be a finite group. We may embed it via the Cayley embedding ...

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359 views

### Which finite nonabelian groups have all their quaternionic representations of degree one?

A finite group $G$ has a finite set of irreducible representations over the complex numbers. All of these representations are linear (that is, are maps in 1x1 complex matrices) if and only if $G$ is ...

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460 views

### A conjecture on solvablity of finite groups

Suppose $G$ is a finite group and $A$ an abelian subgroup. Suppose for some natural number $n\geq 2$, elements of $\gamma_n(G)$ have the form $[a, x]$ where $a\in A$ and $x\in G$. Then $G$ is ...

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997 views

### Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?

Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$.
Can we always find such an irreducible ...

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863 views

### (A very limited instance of) Lagrange's Theorem's converse and A_5

Suppose $G$ is a finite simple group and $|G|$ is a multiple of $60$. Does it follow that $G$ has a subgroup isomorphic to $A_{5}$? If so, can this be proven without using the Classification?

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494 views

### A Perturbation problem for U(n)

Let G be a finite subgroup of U(n), the unitary group acts on $\mathbb{C}^n$. If there is a unit vector $x$ in $\mathbb{C}^n$ such that g(x) is almost orthogonal to x, for all $g\in G$ except the ...

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613 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 ...

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votes

**1**answer

1k views

### Incompressible surfaces in an open subset of R^3

Let $U$ be a connected open subset of $\mathbb R^3$. Furthermore, we have:
$\mathbb R^3\setminus U$ has exactly two connected components (thus by Alexander duality, $H_2(U;\mathbb Z)=\mathbb Z$).
...

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votes

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886 views

### Groups which satisfy Mal'cev's theorem (locally residually finite)

Recall that a group $G$ is residualy finite if for every non-zero element $g\in G$ there exists a homomorphism $\sigma:G\rightarrow H$ such that $H$ is finite and $\sigma(g)\neq 0$. Mal'cev's theorem ...

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**1**answer

367 views

### Perfect group of order 190080

I need to know some properties of the perfect group of order $190080$
which is the Schur cover of the Mathieu group ${\rm M}_{12}$, but
when using ...

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877 views

### Efficient presentations for finite groups

A finitely presented group which has more generators than relations has an infinite abelianization and so is an infinite group. Therefore, for a finite group, all presentations must have at least as ...

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**1**answer

158 views

### 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], ...

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353 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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votes

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533 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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votes

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907 views

### Diameter of universal cover

Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric).
What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or ...

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610 views

### A group action of the Heisenberg group with special symmetries

Suppose we look at the Heisenberg group $H_{d}$ as a matrix group of upper triangular matrices over the ring $\mathbb{Z}/d\mathbb{Z}$. You can even choose $d$ to be prime if you want. A natural ...

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315 views

### 3-cocycle representatives for the dihedral group $D_{2n}$?

I am looking for a reference for a complete list of 3-cocycle representatives for $H^3(D_{2n},\mathbb{C}^\times)$, where
$$
D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle
$$
is the dihedral group of ...

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votes

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348 views

### Cohomology of $T^n/W$ for compact Lie groups

Let $G$ be a compact, connected and simply connected Lie group.
Let $T\subset G$ be a maximal torus and let $W$ be the corresponding
Weyl group.
Then we have the diagonal action of $W$ on $T^{n}$ ...

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votes

**1**answer

347 views

### Do subgroups have “two sided bases”?

Let $H\leq G$ be an inclusion of finite groups. Define a map $E\colon \mathbb{C}[G]\to \mathbb{C}[H]$ to be the $\mathbb{C}$-linear extension of
$$
E(g)=\begin{cases}
g &\text{if } g\in H\\\
0 ...

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votes

**1**answer

387 views

### Factorization of a finite group by two subsets

I want to write a GAP program for checking the following question.
Let $G$ be a given finite group with order $n$. Is it true that for every factorization $n=ab$ there exist subsets $A$ and $B$ ...

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votes

**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 ...