Questions about the branch of abstract algebra that deals with groups.

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4
votes
2answers
291 views

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 ...
2
votes
2answers
181 views

Minimal *-idempotents for the group algebra of the symmetric group

There is a well-known construction of minimal idempotents in the group algebra of the symmetric group $\mathbb C[S_n]$ using row symmetrizers and column antisymmetrizers. But these idempotents are ...
8
votes
3answers
391 views

For which series of finite simple groups is it algorithmically decidable whether they contain a homomorphic image of a given finitely presented group?

Let $G$ be a group given by a finite presentation. On the one hand, it is easy to determine the abelian invariants of $G$, or in other words, it is algorithmically decidable whether $G$ surjects to a ...
5
votes
1answer
356 views

Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split? If not, is there an example? ...
3
votes
1answer
63 views

Is the annihilator of the intersection of two subgroups of a (countable) discrete abelian group generated by the annihilators of the two subgroups?

Let $G$ be a (countable) discrete abelian group and denote by $\hat{G}$ its Pontryagin dual, i.e. the compact abelian group of group homomorphisms $\chi:G \longrightarrow \mathbb{T}$. Recall that, for ...
3
votes
1answer
192 views

Computation of Galois group

I'm studying methods of computation of Galois group of irreducible polynomials over $\mathbb{Q}$. In case of fifth degree there are 5 variants of Galois ...
4
votes
0answers
174 views

Schreier's formula and descending chains

For a group $G$ we denote by $d(G)$ the cardinality of a smallest set of generators. A finitely generated group $G$ is said to satisfy Schreier's formula if for every subgroup $H \subseteq G$ of ...
0
votes
0answers
60 views

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 $g\in G$ ...
5
votes
1answer
116 views

Nearly slender abelian groups

Let $\prod_{n=1}^{\infty}\mathbb{Z}$ be the Baer-Specker group (infinite direct product of the additive group of integers) and $\bigoplus_{n=1}^{\infty}\mathbb{Z}$ be the natural subgroup which is the ...
3
votes
2answers
201 views

(Quadratic) equation in free group?

I know almost nothing about this field, so my following question may be stupid. We know in a free group F, if $ax^{-1}a^{-1}x=1$, then $x=t^{k}, a=t^{l}$ for some t. Now consider a more general ...
1
vote
3answers
264 views

The number of non isomorphic groups in Cext(G,C_p)

Let $p$ be a prime number, $C_p$-cyclic group of order $p$, and $G$ an elementary p-group of order $p^n$. Let us denote by Cext$(G,C_p)$ the group of all central extensions of $C_p$ by $G$. Is the ...
0
votes
1answer
115 views

Known computations of certain 2-cohomology groups?

I wanted to know if there are any computations of cohomology groups $H^n(\Gamma,A^{(\Gamma)})$ in the literature for certain $n\in\mathbb{N}$, Abelian groups $A$, and infinite groups $\Gamma$. Here ...
27
votes
3answers
954 views

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 ...
6
votes
1answer
351 views

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 ...
1
vote
2answers
354 views

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 $G \cong A_5\times A_5$? ...
0
votes
1answer
104 views

Why are all involutions conjugate in the special linear group of degree 2?

It appears to be standard that the set of non-identity involutions in $SL(2, 2^n) = PSL(2, 2^n)$ forms a single conjugacy class. What is the best reference for this? I note that ...
6
votes
1answer
216 views

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 ...
7
votes
3answers
622 views

Is group theory useful in any way to optimization?

For what I have seen, optimization uses a lot of linear algebra and convex analysis, but I have not seen any group theory being used, so I was curious about it. Is group theory useful in any way to ...
1
vote
0answers
47 views

Some resources about minimum-length generator sequences

In the group theory I want to know what are the best results known for problem of finding minimum-length generator sequences. This problam have different titles in articles that cause difficality in ...
2
votes
2answers
137 views

$PSL_2(\mathbb{Z}/p^n)$ isomorphic to automorphism group of depth-$n$, $(p+1)$-regular tree?

A comment on another question (linked below) states "The group $PSL_2((\mathbb{Z}/p^n))$ is the automorphisms group of the $(p+1)$ regular tree of depth $n$, where at level $m$ of the tree you have ...
3
votes
1answer
137 views

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 ...
4
votes
0answers
100 views

Regularity of polynomial growth of groups

Let $G$ be a finitely generated group of polynomial growth. This means that the size $B_n$ of the ball of radius $n$ satsifies: $$ A n^d \leq B_n \leq Bn^d $$ for some constants $A$, $B$. My question ...
23
votes
1answer
705 views

How strong is this conjecture? $(Z/nZ)^*$ is generated by “small” elements

Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$. Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My ...
0
votes
1answer
128 views

A question on direct limits of finite $p$-groups

Where can we find a well developed material on direct limits of finite $p$-groups? For instance, is there a characterization of such groups, which have a finite rank (that is every subgroup can be ...
1
vote
1answer
63 views

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 ...
2
votes
0answers
43 views

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 ...
0
votes
1answer
191 views

Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$

Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...
4
votes
2answers
300 views

a question on rank of fundamental group

Assume $G$ be the fundamental group of a closed orientable hyperbolic 3-manifold. Let $G_{1} = \langle a_{1},...,a_{k} \rangle$ be a free subgroup of $G$, and let $G_{2}=\langle a_{k+1} \rangle$ be a ...
3
votes
2answers
203 views

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 ...
2
votes
0answers
69 views

“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 ...
10
votes
1answer
305 views

Finitely presented group in which every element is conjugate to its square

Does there exist a nontrivial finitely presented group in which every element is conjugate to its square? Is this an open problem? Motivation: Jahren proved in [Geom Dedicata (2010)] that if $M$ is a ...
4
votes
1answer
395 views

Infinite groups containing maximal subgroups that are abelian

If G is a finite group which contains a maximal subgroup M which is abelian, then it is an exercise to show that G is solvable and that the third term in the derived series equals 1. What happens for ...
1
vote
0answers
111 views

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: - ...
4
votes
0answers
88 views

When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?

Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...
9
votes
2answers
306 views

Can the difference of non-conjugate pseudoreflections lie in the commutator subgroup?

Let $G$ be a finite group acting on a complex vector space $V$ by pseudoreflections (i.e. every element of $G$ is a product of elements which fix hyperplanes in $V$). I would like to understand the ...
8
votes
1answer
248 views

Reference for the fact that $SL_n(O_K)$ surjects onto $SL_n(O_K/I)$ for any ideal I

Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map ...
12
votes
3answers
491 views

Is {6,3,7} an 'ultrahyperbolic' Coxeter group?

These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the {6,3,7} Coxeter group, by which I mean the one coming from the Coxeter diagram ...
8
votes
1answer
224 views

Question about doubly transitive groups with an n-cycle

Let $G$ be a doubly transitive subgroup of $S_n$ which contains an $n$-cycle, and let $G_{12}$ be the subgroup of $G$ consisting of all elements $g\in G$ for which $g(1)=1$ and $g(2)=2$. Then ...
0
votes
2answers
180 views

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 ...
4
votes
0answers
137 views

Is there a notion of “tame” representations of $GL_n(Z)$?

This is a followup to this question about the (left) noetherianity of the group ring of $GL_n(\mathbf{Z})$: Does GL_n(Z) have a noetherian group ring? Given that $\mathbf{Z}[GL_n(\mathbf{Z})]$ is ...
14
votes
1answer
486 views

Does GL_n(Z) have a noetherian group ring?

Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL_n(Z)$ been considered in the literature? Motivation: a recent trend has been to study ...
10
votes
0answers
233 views

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 ...
48
votes
3answers
2k views

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 ...
6
votes
2answers
311 views

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 $S_{n}$?
3
votes
2answers
126 views

All Integers from the Smallest Digit Stream with a Window Filter

Let's represent integers with D digits where each digit has B values (i.e., the base is B and we effectively work only with integers between 1 and B^D). Is it possible to choose a single ...
4
votes
2answers
533 views

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 ...
7
votes
1answer
385 views

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 ...
4
votes
1answer
164 views

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.
2
votes
0answers
84 views

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 ...
2
votes
1answer
320 views

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 ...