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

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11
votes
0answers
142 views

What Is The Minimal Monomial of the Symmetric Group?

In the symmetric group $S_n$ what is the shortest sequence $c_1,\ldots,c_k\in S_n$ such that, for all $x\in S_n$ the following product of conjugates of $x$: $$x^{c_1}x^{c_2}\ldots x^{c_k}$$ equals the ...
3
votes
1answer
141 views

the number of indecomposable modules of finite groups over finite fields of a fixed dimension

I am interested in determining the the number of indecomposable modules of finite groups over finite fields of a fixed dimension. Specifically, I have the following conjecture: Conjecture. Suppose we ...
3
votes
1answer
182 views

Special linear groups over function fields

Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F_q$ the unique field with $q$ elements. What is known about the structure and properties of $\mathrm{SL}_2(F_q[t])$ as ...
11
votes
0answers
199 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 ...
3
votes
0answers
106 views

Invariant Laurent polynomials under cyclic group action

Start with the cyclic group $G:=\mathbb{Z}/p$ of prime order $p$ and and an integer lattice $P:=\mathbb{Z}^p$. Let $G$ act on $P$ by cyclic permutation of coordinates. There is an induced action on ...
3
votes
2answers
195 views

Limits of conjugated subgroups

I've recently encountered the following problem. Given a group $G$, a subgroup $H$ and a sequence $g_n\in G$, let $$ \liminf_{j\to\infty}H^{g_j} :=\bigcup_{n\ge 1} \bigcap_{j\ge n} H^{g_j}.$$ Here $$ ...
3
votes
0answers
108 views

Equalizer in Free groups

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and for each $1\leq i\leq n$, let a homomorphism $d_i:F_n\to F_{n-1}$ be defined as follows: $d_i(x_r)=x_r$, if $i>r$; $d_i(x_r)=1$, if ...
11
votes
1answer
348 views

Is there a faithful transitive locally finite action of the modular group?

Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?
0
votes
0answers
64 views

When is edge colored circulant isomorphism polynomial?

Don't understand enough group theory, but two papers appear to give partial results about an open problem. Edge colored graph isomorphism is isomorphism which preserves the edge coloring (the ...
1
vote
0answers
128 views

Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
5
votes
1answer
355 views

Why is this group called “The Holomorph of a group”

Many years ago I found in google the notation "Holomorph of group". It is the semi direct product of $G$ with $Aut(G)$. Why is the term "Holomorph" used here, while it is usually used for complex ...
1
vote
0answers
76 views

Are lattices in the special real linear group subgroup seperable?

Let $G \leq SL_2(\mathbb{R})$ be a lattice, let $H \leq G$ be a finitely generated subgroup of infinite index, and let $n \in \mathbb{N}$. Must there be some $H \leq U \leq G$ such that $n \leq [G : ...
6
votes
0answers
156 views

An example of a simple infinite 2-group

I've asked this question before on Mathematics, and they suggested me to ask here (Link). Is there an example of a simple infinite $2$-group? Informations If a $2$-group is Artinian I know that ...
6
votes
3answers
1k views

A categorical method to, say, determine the cardinality of a group

I am trying to figure out how much one can figure out about an object using category theory. Ideally, any property that is well defined up to isomorphism should be computable using only category ...
5
votes
2answers
222 views

Embedding $G$ in a $Z(G)$ extension of $\operatorname{Aut}G$

This question follows up a question I asked on math.SE. This is a refinement and a reference request. For what groups $G$ does there exist a $Z(G)$-extension of $\operatorname{Aut}G$ (call it ...
2
votes
2answers
210 views

F.g group with infinite ends not Q.I to a free group

Is there any easy example of a finitely generated group with a Cantor set of ends that is not quasi-isometric to a finitely generated free group? Thanks in advance.
2
votes
0answers
117 views

Special sets of involutions generating ${\rm S}_n$

For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$ $(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?: for any $n$, the $g_{n,i}$ $(i = 1, ...
2
votes
1answer
201 views

twists of algebraic groups

If $k$ is some field - for convenience, of characteristic 0 -, $\bar{k}$ is an alg. closure of $k$, and $G$ is some $k$-algebraic group, one can define a twist of $G$ to be some $k$-algebraic group ...
1
vote
0answers
33 views

About Abelian Radicable Groups Generated by Chernikov's subgroups

Let $G$ be a abelian and radicable group generated by subgroups abelian, radicable and Chernikov. Then $G$ is Chernikov? ie, if $G = \left<H| H \leq G \right>$ with $H$ abelian, radicable and ...
4
votes
0answers
267 views

Very frustrated reading a proof of the faithfulness of Artin's representation of braid groups

I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper: (The proof is not finished yet but I am very confused by now.) ...
4
votes
1answer
227 views

A finite 2-group acts on an elementary abelian group of odd order

I would be grateful if you have an idea how to prove the following: Let a finite $2$-group $P$ act on an elementary abelian group of odd order $N$ such that the centralizer $C_P(N)=1$, and then form ...
8
votes
1answer
234 views

Condition for a certain subset being a subgroup

For any finite group $G$ and $n$ a divisor of $|G|$, consider the following subset of elements of "co-order" dividing $n$: $$G(n) = \{ g \in G \mid g^{|G|/n} = 1 \}$$ By a classical theorem of ...
2
votes
0answers
74 views

Orthosymplectic group, matrix representations

We have the orthosymplectic $osp(n,m|2k)$. The bosonic part is $so(n,m)\times sp(2k)$. The lie algebra generators are given in eg http://cds.cern.ch/record/524737/files/0110257.pdf$ where the group ...
30
votes
1answer
593 views

Two groups that are the automorphism groups of each other

Let $H,K$ be two non-isomorphic groups such that $H\cong Aut(K)$ and $K\cong Aut(H)$. Is there any example of such groups ? Note: I had asked the question there.
0
votes
0answers
59 views

adjoint quotient and points in DVRs

Let $G$ be a connected reductive group over an algebraically closed field $k$, $T$ a maximal torus and $W$ its Weyl group. We have a Steinberg map $\chi:G\rightarrow \mathfrak{C}:=T/W$ if we have a ...
4
votes
0answers
81 views

Connected compact Lie groups with Lie algebra so(4n, R)

I am trying to write a complete list of connected compact simple Lie groups (or of connected complex simple Lie groups, both tasks are equivalent). I am missing just one case. Consider the Lie ...
2
votes
1answer
113 views

Totally aperiodic sequence

Let $A$ be a finite set. Let $A^k$ be the set of words in the alphabet $A$ of length $k$ and $A^*$ be the set of infinite words. I was looking for an element $a = \lbrace a_n \rbrace_{n \in ...
3
votes
1answer
119 views

Schreier's formula and supersolvable groups

A finitely generated profinite group $G$ is said to satisfy Schreier's formula if for every open subgroup $L \leq_o G$ we have $d(L) = (d(G)-1)[G:L] + 1$. Here $d$ stands for the smallest cardinality ...
1
vote
0answers
100 views

Can a profinite completion be free pro-p?

Is there a prime number $p$ and a finitely generated residually finite group whose profinite completion is a free pro-$p$ group on a nonempty finite set? Thanks to YCor we see that we cannot take the ...
2
votes
1answer
186 views

$nse$ for which simple group was determined?

Let $G$ be a finite group and $\omega(G)$ be the set of element orders of $G$. Let $k\in\omega(G)$ and $m_k$ be the number of elements of order $k$ in $G$. Set $nse(G):= \{m_k : k \in\omega(G)\}$. ...
11
votes
3answers
920 views

Non-abelian Grothendieck group

By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations ...
0
votes
0answers
111 views

What does the ring $K[S]$ know about a group generated by $S$?

Given a discrete group $\Gamma$ generated by $S$ let $K[S]$ denote the subring of the group-ring $K[\Gamma]$ generated by $S$ (over a commutative ring $K$). The ring $K[S]$ is thus a quotient of the ...
3
votes
1answer
182 views

Can the full and reduced group $C^*$-algebras be “noncanonically” isomorphic?

Is there a locally compact group $G$ such that the canonical map from $C^{*}(G)$ to $C^{*}_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{*}$ algebras are isomorphic ...
3
votes
1answer
193 views

Are there quasiconvex normal subgroups?

Let $G$ by a hyperbolic group, and let $H \lhd G$ be a normal quasiconvex subgroup. Is it possible that $|H| = [G : H] = \infty$ ?
2
votes
1answer
105 views

Conjugates and infinite index subgroups of free groups

Here I am asking for an analogue of Generating infinite index subgroups of a free group Let $F$ be a nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of ...
2
votes
0answers
158 views

Exotic actions of hyperbolic groups

Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that: The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$. For every quasiconvex subgroup $H \leq ...
7
votes
0answers
141 views

Is there an infinite increasing sequence of naturals for which Landau's function can be efficiently computed?

Landau's function $g(n)$ is the largest order of an element of the symmetric group $S_n$. Equivalently, $g(n)$ is the largest least common multiple (lcm) of any partition of $n$. In general $g(n)$ is ...
1
vote
1answer
165 views

Reconstructibility of a group from subgroups

Let $G,H$ be groups and suppose that $\varphi: G\to H$ is a bijection such that for any proper subgroup $G'\neq G$ of $G$ the image $\varphi(G')$ is a subgroup of $H$ and the restriction ...
0
votes
1answer
228 views

Number of involutions in a finite group [closed]

Let $G$ be a finite group. Does it possible to determine number of involutions in it? If not, is there any bound for it?
1
vote
0answers
69 views

Normal p-complement and the Frattini subgroup [closed]

A normal subgroup $N$ of a finite group $G$ is said to be a normal $p$-complement if $(p,|N|)=1$ and there is a Sylow $p$-subgroup $P$ of $G$ such that $G=NP$. The definition of a normal ...
3
votes
1answer
168 views

How to think about the simple reflection s_0 in the affine Weyl group?

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
3
votes
2answers
144 views

Solubility of an algebraic group and Weil restriction

For $\ell/k$ a finite separable extension of fields and an affine variety $X_{/\ell}$, the Weil restriction of scalars $R_{\ell/k}(X_{/\ell})$ represents the functor $R_{\ell/k}(X_{/\ell})(S) := ...
5
votes
1answer
209 views

A hyperbolic group with a small profinite completion

Is there a finitely generated non-elementary word hyperbolic group the profinite completion of which is known (or conjectured) to be rather restricted, that is: abelian, pro-$p$, virtually ...
6
votes
1answer
284 views

Are the distributive permutation groups linearly primitive?

An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$. It is called primitive if it is transitive and preserves no ...
6
votes
2answers
303 views

Is residual finiteness a property of “many” finitely presented groups?

Is there a reasonable random model for selecting a finitely presented group $G$ such that with positive probablity (or even with probability almost $1$) some of the following properties hold: $G$ is ...
6
votes
1answer
148 views

Is there a highly transitive action of a finitely generated torsion simple group?

Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ? Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two ...
1
vote
1answer
141 views

Does the hyperoctahedral group have only 3 maximal normal subgroups?

An hyperoctahedral group $G$ is the wreath product of $S_2$ and $S_n$, where $S_{n}$ is the symmetric group on $n$ letters, or in other words the semi-direct product $G=S_2^n\rtimes S_n$, w.r.t. the ...
5
votes
1answer
174 views

What is the Schur multiplier of the affine linear group AGL(n,q)?

What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements? I am particularly interested in the simple case $n=1$. Computation ...
4
votes
1answer
252 views

Generating infinite index subgroups of a free group

Let $F$ be nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of infinite index and let $x,y \in F \setminus H$. Must there be some $a \in F$ such that $[F : ...
7
votes
3answers
245 views

Polarizations generate the ring of invariants?

The symmetric group $S_n$ acts on $\mathbb R^n$ by permuting the coordinates and the ring of polynomial invariants is generated by the elementary symmetric polynomials. If we restrict the action to ...