The finite-groups tag has no wiki summary.

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### Making a profinite group free

Let $F$ be a free profinite group, $G$ a profinite group. Suppose that the free profinite product $F \amalg G$ is a free profinite group. Must $G$ be a free profinite group?
For abstract groups the ...

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### groups of order $ p(p^2-1) / 4 $ where $p$ is a prime

Let $p> 3$ be a prime number and $G$ be a finite group of order $p(p^2-1) / 4 $. If $ 4 \mid (p+1)$ then easily we can see that the $ p$-Sylow subgroup of $G$ is a normal subgroup of $G$. As I ...

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### Union of conjugates of a subgroup

Let $G$ be a finite group, $H \leq G$ a proper subgroup. It is well known that the union of the conjugates of $H$ does not cover $G$. I would like to know of more precise results (even in special ...

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### smooth quotient out of a singular variety?

If $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $G$ is a finite group acting faithfully on $X$, then the Shepard-Todd theorem gives us some criterion for $X/G$ to be smooth.
My ...

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### What is natural about the well-known bijection between conjugacy classes and irreps of a symmetric group?

Symmetric groups possess a well-known bijection between conjugacy classes and irreducible representations. More precisely, both sets are indexed by Young diagrams.
Question: To what extent is this ...

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### maximal subgroup of a $p$-group

Let $G$ be a $p$-group of order $p^t$, and also let $exp(G)=p$. Is there such a group $G$, such that every maximal subgroups of it, not being abelian? In fact, I want an example of such a group with ...

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### Subgroup property stronger than being characteristic

In what follows, all groups are assumed to be finite.
Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a weakly closed subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ ...

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### On the Suzuki group

Let $G$ be the Suzuki group over the field with $q=2^{2m+1}$ elements, $m>0$. Then, by Theorem 3.10 from B. Huppert, N. Blackburn, Finite Group III, pp 192-193, or wikipedia, the group $G$ contains ...

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### Generators of Sylow subgroups

Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$?
Here $d(H)$ denotes the ...

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### Enumeration of a finite group

Let $G=\{g_1,g_2,...,g_n\}$ be a group with $e=g_1$ and $n$ is odd,
Set $$a_1=g_1$$
$$a_2=g_1g_2$$
$$a_3=g_1g_2g_3$$
$$a_n=g_1g_2...g_n$$
I am looking for example that all $a_i$ are different from ...

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### 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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### Finite Quotients and Resolutions of Singularities

So, I feel like I'm missing something obvious, but I have the following situation:
Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to ...

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### A semisimple group ring

Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple?
As noted ...

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### 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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### Finite groups factorized into two simple alternating groups

My research is somehow related to the following question :
Describe and classify all finite groups $G$ such that $G=HK$ with $H \cap K=1$, where $H \cong A_m$ and $K \cong A_n$ for some integers $m, ...

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### Integral representations of groups of small order

I have a problem in which it would be helpful to know about the integral representations of some groups of small order (probably of fairly low degree). From what I've gathered so far, cyclic groups of ...

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### What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?

Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}_q$.
The Grothendieck group of the category of ...

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### Number of generators of the commutator

Can one find a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for every finite supersolvable group $G$ we have: $d(G') \leq f(d(G))$?
Here $d(K)$ is the cardinality of a minimal set of ...

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### Eigenvalues of the Cayley-like graph

Let $ F_q $ be a finite field of characteristic 2.
Let $ x^2 + Sx +P \in F_q[x] $ be an irreducible polynomial over $ F_q $,
and let $ g $ be one of its roots in $ F_{q^2} $.
Define a map $ M: ...

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### Elementary abelian $p$-subgroups of maximal rank in finite groups of Lie type

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a reductive group defined over $\mathbb{F}_p$. For any $d\in\mathbb{Z}^+$, let $C_d(G)$ be the set of conjugacy ...

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### Maximal order of finite subgroups of $GL(n,Z)$

I am interested in the finite subgroups of $GL(n,Z)$ of maximal order.
Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed ...

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### 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 ...

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### A question on the poset of classes of isomorphic subgroups of finite groups

Given a finite group $G$, we consider the set $${\rm Iso}(G)=\{[H]\mid H\leq G\},$$where
$[H]=\{K\leq G\mid K\cong H\}, \forall H\leq G$. Then ${\rm Iso}(G)$ can be partially ordered by defining
...

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### Profinite groups, completions, and Schreier's formula

Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...

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### Profinite completions

I call a profinite group $G$ Noetherian, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is ...

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### a characterization for cyclic groups [duplicate]

Let $G$ be a group of order $n$ and its subgroup lattice be order-isomorphic to that of $\Bbb Z_n$. Is $G$ cyclic?

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### Fibres of square map on finite groups and the inverse set of a subset

Let $G$ be a finite group and $q:G\rightarrow G:g\mapsto g^2$ the square map. Now, if $A$ is a subset of one fibre of $q$, i. e.
$$a^2=b^2$$
holds for all $a,b\in A$, is there always some $g\in G$ ...

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### When is $S_n \times S_m$ a subgroup of $S_p$?

I asked the following question on math.stackexchange several months ago:
Let $n,m,p>1$ be such that $S_n \times S_m \hookrightarrow S_p$. Does it imply that $p \geq n+m$?
Derek Holt gave a ...

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### Paths in groups

Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$.
Question: For what groups does there exist a Hamiltonian ...

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### on the extensions of $ A_5$ by $A_5$ [closed]

Let $G$ be a finite group such that $G$ has a normal subgroup $H$ and $H$ is isomorphic to the alternating group $A_5$. Also we know that $G/H \cong A_5$.
Can we say that $G \cong A_5\times A_5$?
...

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### Fast construction of straight line programs?

Given a group $G$ and a set of generators $A$, we can ask ourselves (and do ask ourselves all the time) to bound the diameter of $G$ with respect to $A$. The diameter, let us recall, is defined to be ...

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### Aschbacher's description of Sylow 2-subgroups

I'm looking for a few more examples of using Aschbacher (1980)'s “fundamental SL2 subgroups” description of Sylow 2-subgroups of finite groups of Lie type in odd characteristic. I do not yet ...

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### A characterization of almost relatively free, finite $p$-groups

Let $G$ be a finite minimally $d$-generated $p$-group.
If $G$ is relatively free, that is $G$ is a quotient of the free group $F$ on $d$ generators by a fully invariant subgroup of $F$, then the ...

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### A question on $p$-central $p$-groups

Let $p$ be a fixed prime. A group $G$ is termed $p$-central if every element of order $p$ in $G$ lies in the center.
Having a finite $p$-group $G$ of rank $k$ (the least integer, such that every ...

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### On the groups of order $p(p^2+1)$

Let $G$ be a group of order $p(p^2+1)$, where $p$ is an odd prime number and $p>3$. Easily we can see that $G$ is solvable and so $G$ has a Hall subgroup $L$ of order $p^2+1$. Also we know that ...

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### “A locally dual polar space for the Monster”

I am currently looking at Ronan and Stroth's 1984 paper Minimal Parabolic Geometries for the Sporadic Groups. When considering the $3$-minimal parabolic system of $F_{1}$, they cite a preprint by ...

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### Questions about some finite p-groups of coclass 2

I have two questions about a specific type of finite $p$-groups that i've seen in an interesting work on automorphisms of finite p-groups.
Let $G$ a finite $p$-group of order $p^{n}$ such that:
- ...

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### Projective characters with corresponding factor set

The following is just a follow up to my previous question. I have a finite group $H$ with 14 ordinary characters. The Schur multiplier $M(H)\cong 2^2$. Hence the group $H$ will have 3 sets of ...

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### Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability ...

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### How do I verify the Coq proof of Feit-Thompson?

I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is ...

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### Sylow 3-subgroups of symmetric group

Is there any routine technique to find a set of permutations which generate a Sylow $3$-subgroup of the symmetric group $S_{n}$?

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### Decomposing representations of finite groups

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.
Must there be ...

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### Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...

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### Maximum length of chains of subgroup in GL(n,q)

Let G=GL(n,q) be a general linear group n-dimensional over a field with q element (q power of a prime). I am looking for an estimate of maximum length of chains of subgroup in G.
Thanks.

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### Coprime automorphisms of finitely generated pro-$p$ groups

Let $P$ be a finitely generated pro-$p$ group and let $G$ be a semidirect product $P \rtimes A$, where $A$ is a finite group of order coprime to $p$ that acts faithfully on $P$. Then one can show ...

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### Groups like symmetric group

For sufficiently large n consider this question
Let $G$ be a finite group with the following properties:
$|G|=n!$
$H,K$ are subgroups of $G$ such that $H\cap K=1$ and $H\cong S_{3}$ and $K\cong ...

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### Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?

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### generality of the lattice of normal subgroups

Let $(X,\le)$ a (finite) modular lattice. Is there a (finite) group $G$ such that the lattice of all normal subgroups of $G$ is isomorphic to $(X,\le)$?

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### Which finite groups can be characterized by their subgroup orders?

Given a finite group $G$, we denote by $\pi_s(G)$ the set of orders of its subgroups. Which finite groups $G$ can be characterized by the set $\pi_s(G)$, i.e. $\pi_s(H)=\pi_s(G)$ implies $H\cong G$? ...

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### Subgroups from which all class functions extend to class functions on the ambient group

Until someone suggests better terminology, let me call a subgroup H of a finite group G segregated if every class function on H can be extended to a class function on G. Equivalently, H should have ...