Questions on group theory which concern finite groups.

**37**

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### Normalizers in symmetric groups

Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$.
...

**30**

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553 views

### Class function counting solutions of equation in finite group: when is it a virtual character?

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by
$$ f_w(g) = |\{ (x_1,\dots, x_n)\in G^n\mid ...

**22**

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475 views

### Next steps on formal proof of classification of finite simple groups

While people are steaming ahead on finessing the proof of the classification of finite simple groups (CFSG), we have a formal proof in Coq of one of the first major components: the Feit-Thompson ...

**21**

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272 views

### Guess that group via product queries

Suppose someone (person B) knows a finite group $G$ of order $n$.
You (person A) know only the order $n$,
and that $1$ is the name of the identity element.
The group elements are named $1,2,\ldots,n$ ...

**19**

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616 views

### Is there a Mathieu groupoid M_31?

I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...

**19**

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445 views

### Density of first-order definable sets in a directed union of finite groups

This is a generalization of the following question by John Wiltshire-Gordon.
Consider an inductive family of finite groups:
$$
G_0 \hookrightarrow G_1 \hookrightarrow \ldots \hookrightarrow G_i ...

**18**

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761 views

### Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...

**16**

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375 views

### Applications of the surjectivity of Brauer's decomposition map over arbitrary fields?

Recently I've been going over some of Serre's reformulation of Brauer theory with a student, following the influential treatment in Part III of Serre's lectures (revised 1971 French edition) later ...

**15**

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303 views

### Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...

**14**

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568 views

### What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in
Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11,
London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...

**13**

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622 views

### How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?

This is a follow-up to a recent mathoverflow question
34387
about computing the orders of finite unitary groups and the comments made there.
Between 1955 (Chevalley's Tohoku paper) and 1968 ...

**12**

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651 views

### What can be said about Schur indices, given only the character table?

Let $\chi$ be an irreducible (complex) character of a finite group, $G$. The Schur index $m_{K}(\chi)$ of $\chi$ over the field $K$ is the smallest positive integer $m$ such that $m\chi$ is afforded ...

**11**

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281 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 ...

**10**

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271 views

### Solving a set of equations in a finite symmetric group

A standard way to find solutions to a finite set of equations in a finite symmetric group
${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use
the low index subgroups ...

**10**

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357 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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159 views

### Is an inclusion of finite groups with boolean lattice, linearly primitive?

Let $(H \subset G)$ be an inclusion of finite groups.
Definition: Let $W$ be a representation of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{H}:=\{w \in W \ \vert \ kw=w \ ...

**9**

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351 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 ...

**9**

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465 views

### How many sporadic simple groups are there, really?

I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ...

**8**

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81 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 ...

**8**

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157 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 ...

**8**

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283 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, ...

**7**

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120 views

### Rings that are $K_0$ of finite groups

Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of ...

**7**

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208 views

### A strong sum-product “for translates” in finite fields

In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...

**7**

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134 views

### Fusion pattern in a cyclic subgroup of order 8

Can a finite simple group $G$ have an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$?
In other words, does this fusion ...

**7**

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338 views

### Outer group automorphisms preserving conjugacy classes of pairs of commuting elements

The following problem appears in the study of braided autoequivalences of representation categories of Drinfeld doubles of finite groups.
Let $G$ be a finite group. An outer automorphism $\alpha$ of ...

**7**

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232 views

### Intersections of anisotropic tori with split Levi subgroups

Let G be a connected reductive group defined and split over a finite field k with Frobenius morphism F. Let T be an F-stable minisotropic maximal torus. Let P be an F-stable proper parabolic ...

**7**

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499 views

### Cyclic Sylow $p$-subgroup in finite simple groups

Hello,
I came accross a beautiful and "simple" (no pun intended) theorem, mentioned here in slide 14 by Jack Schmidt.
All finite simple groups have a cyclic
Sylow $p$-subgroup for some $p$
I ...

**7**

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243 views

### Why would dim primitive irrep divide size of some conjugacy class ?

From Isaacs et.al. 2005
Conjecture C. Let χ be a primitive
irreducible character of an arbitrary
ﬁnite group G. Then χ(1) divides |
clG(g)| for some element g ∈ G.
Here, of course, we ...

**7**

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353 views

### The maximal order of an element in orthogonal groups over finite fields of characteristic 2

Let $q$ be a power of $2$ and let $(V,Q)$ be a
quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type ...

**6**

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209 views

### Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...

**6**

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239 views

### Recovering a group from its number of symmetric embeddings?

Fix a finite group $G$, and let $f_G:\mathbb{N}\rightarrow\mathbb{N}$ be defined by setting $f_G(n)$ to be the largest $m$ such that $S_n$ contains $m$ disjoint (pairwise-intersecting at 1) copies of ...

**6**

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150 views

### Finding a database of representations as matrices

Sorry if this would be more appropriate as a stackoverflow and not a mathoverflow question, but I think it's more likely to be known in this community.
There are plenty of places on the internet or ...

**6**

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215 views

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

Let $G$ be a compact, connected and simply connected.
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 $T^{n}$ for $n\ge ...

**6**

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468 views

### Jordan's Theorem on primitive permutation groups

The theorem I am referring to in the title is this:
Theorem. If $p$ is a prime and $n$ is an integer with $n \geq p+3$, then the only primitive permutation groups on $n$ points containing a ...

**6**

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494 views

### When is Hom(G, H) the same size as Hom(H, G)?

Let $G$ and $H$ be finite groups. Consider the ratio
$$r_{G, H} \equiv {|Hom(G, H)| \over{|Hom(H,G)|}}$$
My question is
When is $r_{G, H} = 1$? Can we characterize the pairs of groups $(G, H)$ ...

**5**

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178 views

### Derived subgroup of rational points versus rational points of derived subgroup

Let $\mathbf G$ be a connected algebraic group defined over a field $\mathbb F_p$. If $q=p^n$, then the groups $\mathbf G^\prime (\mathbb F_q)$ and $\mathbf G (\mathbb F_q)^\prime$ are not always ...

**5**

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99 views

### Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups.
Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups.
Definition: ...

**5**

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84 views

### representable functions on FinGrp

We look at functions $f: \text{FinGrp} \rightarrow \mathbb{N}$ such that $f$ is constant on isomorphism classes.
Let's say that $f$ is representable if there is a (possibly infinite) group $K$ such ...

**5**

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229 views

### A particular example of solvable group

Let be $m$ an integer, $m>0$, $n=p_1^{a_1}...p_t^{a_t}$. Define $f(n)=a_1+...+a_t$.
Let $G$ be a finite group and define $f(G)=\max\left\{\,f(\lvert g\rvert):g\in G\,\right\}$.
Is there a finite ...

**5**

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115 views

### Information about permutation character from local action

Let $G$ be a finite permutation group acting transitively, but not regularly, on a set $V$. Let $H$ be the stabilizer of some point $v\in V$, and suppose that $H$ acts 2-transitively on one of its ...

**5**

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260 views

### Vanishing sums of powers modulo n

Is it possible to describe all positive integer sequences $\{x_i\}$ such that
$$
\sum_i x_i=n
\quad
\hbox{and}
\quad
\sum_i x_i^k\equiv 0\pmod n
\quad
\hbox{for all $k$ (and a given $n$)?}
$$
...

**5**

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150 views

### Subgroups of M-groups

Let $G$ be an M-group. Let $\chi$ be an irreducible character of $G$ and set $$A_{\chi} =
\{ H\leq G\ |\ \exists \psi\in\textrm{Lin}(H): \psi^G = \chi\}$$
Since $G$ is an M-group, this set is ...

**5**

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350 views

### What are the relation between Rep(G) and Rep(S_n)?

Let G be a finite group. We know it can be written as a subgroup of S_n. On the other hand, people sometimes say Rep(G) --- the category of all finite dimensional representations, are more interesting ...

**4**

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99 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 ...

**4**

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238 views

### Euler characteristic, character of group representation and Riemann Roch theorem

I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in ...

**4**

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216 views

### a characterization for special linear group

Thanks for any help or comments.
In my research I need a characterization for special linear groups over finite fields by some information of its subgroups, especially centralizers. I saw the ...

**4**

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199 views

### Non averaging sequences in finite groups

Let us say that a non averaging sequence in a group $G$ is a sequence $x_1, \dots, x_n$ such that
$$x_i^2 \neq x_j x_k$$
for any indices $i, j, k$ such that two at least are distinct. My question is: ...

**4**

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191 views

### A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a ...

**4**

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59 views

### Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...

**4**

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99 views

### Improvements of the Reidemeister-Schreier index formula for particular classes of groups

I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then
$$d(H) \le (d-1) ...