Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

Filter by
Sorted by
Tagged with
3 votes
1 answer
313 views

Probability in $GL_2(\mathbb{Z}/p^{r}\mathbb{Z})$

My question may be not interesting or easy to answer ! but I am really not familiar with proba. Let $p$ be an odd prime number. and let $r\geq1$ an integer. choose an element $A\in\mathrm{GL}_2(\...
Zakariae.B's user avatar
8 votes
1 answer
350 views

$E_n(\ell^\infty)=SL_n(\ell^\infty)$?

Let $R$ be a commutative unital ring $R$ with unit element $1$. For $n\in \mathbb{N}=\{1,2,3,\cdots\}$, let $SL_n(R)$ be the group of all $n\times n$ matrices with entries from $R$ having ...
KevinC's user avatar
  • 81
4 votes
0 answers
167 views

from 2-cocycle to classifying map

Let $A,E,G:\mathrm{Set}_*\to\mathrm{Grp}_*$ be functors from pointed sets to (discrete) groups ($*=1$) together with natural transformations $i:A\to E, \ p: E\to G$ such that for any set $X$ \begin{...
hsldfgh sfkdlhguh's user avatar
3 votes
0 answers
60 views

Lower Wielandt Series of a finite group

Let $G$ be a finite group. The Wielandt subgroup of $G$ is defined to be the intersection of all the normalizers of the subnormal subgroups of $G$ i.e. $$w(G) = \bigcap_{H \triangleleft \triangleleft ...
R Maharaj's user avatar
  • 366
2 votes
0 answers
256 views

Is conjugacy problem hard in braid group?

Recently I studied the braid group and conjugacy problem. It is believed that conjugacy problem is hard on braid group. My friend gave me an EXE file, and I use it for solving conjugacy problem, as an ...
Meysam Ghahramani's user avatar
7 votes
1 answer
710 views

Characterization of group characters

I wonder how to characterize the characters of a (say, finite) group $G$ as special class functions, in particular for the case $G=S_n$ (symmetric group). The answer to this is presumably well known ...
Gandalf Lechner's user avatar
28 votes
0 answers
657 views

Mathieu group $M_{23}$ as an algebraic group via additive polynomials

An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
Peter Mueller's user avatar
2 votes
1 answer
143 views

Right socle of a group ring

Let $p$ be a prime number and $n$ a positive integer. I want to know what is the (right) socle of the group ring $A=\mathbb Z_{(p)}C_n$, where $\mathbb Z_{(p)}$ is the localization of integers at the ...
karparvar's user avatar
  • 323
6 votes
1 answer
372 views

One-ended finitely presented subgroups of hyperbolic groups

In Hyperbolic groups (page 82), Gromov claims that, by a standard application of Thurston's method of geodesic (hyperbolic) simplices, it can be prove that a hyperbolic group contains finitely many ...
Seirios's user avatar
  • 2,361
4 votes
2 answers
444 views

Number of torsion-free abelian groups

Let $\mathfrak{c}$ be the cardinality of the continuum. How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of ...
Bruce Blackadar's user avatar
2 votes
1 answer
183 views

Number of homomorphism, or number of solution to equations, in finite groups

Let $G$ be a finite group, and let $P$ be a finitely generated group. Consider the number $$n=\#Hom_{Grp}(P,G).$$ It is known (see Number of solutions to equations in finite groups) that under ...
Ehud Meir's user avatar
  • 4,969
1 vote
2 answers
704 views

Poisson Furstenberg Boundary of topological groups, reference request

I'm trying to understand the relations between between the following group properties, in the case of (say, compactly generated locally compact) topological groups: Group growth. Amenability. Poisson ...
Snoop Catt's user avatar
6 votes
1 answer
410 views

What is known about the functor $G\mapsto k[G]^\times/k^\times$?

Let $k$ be a ring (resp. profinite ring), $G$ a group (resp. profinite group), and $k[G]$ the group algebra (resp. completed group algebra). For any such $G$, we may associate to it the group of ...
Will Chen's user avatar
  • 10k
3 votes
0 answers
113 views

The group of automorphisms and anti-automorphisms of the first Weyl algebra

Let $k$ be a field of characteristic zero, and let $A_1=A_1(k)$ be the first Weyl algebra. It is well known (first proved by Dixmier, if I am not wrong) that the group of automorphisms of $A_1$, ...
user237522's user avatar
  • 2,783
2 votes
0 answers
88 views

A question on simplicial groups

Does the adjunction $||:sSet\leftrightarrow Top :Sing$ behave well with respect to simplicial and topological groups in the sense that the unit and the counit maps are maps of group objecs in both ...
user95770's user avatar
  • 143
4 votes
1 answer
166 views

Milnor-Wolf theorem for topological groups

The Milnor-Wolf theorem states that any finitely generated solvable group has either polynomial or exponential growth. Is there an analogous result for locally compact compactly generated groups? ...
Snoop Catt's user avatar
6 votes
2 answers
207 views

groupring morphisms and bialgebra

Let $G_{1}$ and $G_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G_{1}]\rightarrow \mathbb{Z}[G_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G_{...
Ofra's user avatar
  • 1,603
36 votes
0 answers
925 views

Are there infinite versions of sporadic groups?

The classification of finite simple groups states roughly that every non-abelian finite simple group is either alternating, a group of Lie type, or a sporadic group. For each of the groups of Lie ...
Myself's user avatar
  • 576
4 votes
0 answers
182 views

Gaussian actions with no Bernoulli part

In an unrelated research project I came upon an example of a mixing unitary representation $\pi: \mathbb{F}_{\infty}\to B(\mathsf{H})$ of the free group on infinitely many generators, such that no ...
Mateusz Wasilewski's user avatar
11 votes
1 answer
888 views

Normal closures of finitely generated subgroups of a free group

Is it true that for every finitelty generated subgroup $H$ of infinite index in a free group $F$ on the two letters $\{x,y\}$, there exists a finite index subgroup $K$ of $H$, such that the normal ...
Pablo's user avatar
  • 11.2k
0 votes
1 answer
134 views

about maximal subgroup of p-groups [closed]

Thanks for any help or comments. Suppose $G$ is a meta cyclic p-group, i.e. $G$ is an extension of cyclic by cyclic group, Is it true that every nonabelian maximal subgroup of $G$ is meta cyclic?
Maryam's user avatar
  • 99
2 votes
0 answers
93 views

Change of generators and shortest product in groups

Let $G$ be a finitely generated group. For a set of generators $B$ of $G$, $\ell_B(x)$ is the length of the smallest sequence of elements(and inverse of the elements) in $B$, such that the product ...
Chao Xu's user avatar
  • 583
4 votes
0 answers
426 views

Weyl groups of Levi factors and their cosets

Let $G$ be a connected semisimple algebraic group defined over an algebraically closed field $F$ of characteristic $0$. Let $B$ be a minimal parabolic subgroup of $G$ (i.e. a Borel subgroup), let $P$ ...
Sarah's user avatar
  • 41
3 votes
1 answer
220 views

On some classical groups with small permutation degree

The following question arises while I am reading a paper of B. N. Cooperstein. In the Table 1 of his paper, two groups have smaller permutation degree compared to other members in their family: $\...
Y. Zhao's user avatar
  • 3,317
9 votes
2 answers
748 views

Solutions of $x^d=1$ in the symmetric group

L Moser and M Wyman, On solutions of $x^d = 1$ in symmetric groups, Canad. J. Math., 7 (1955), pages 159-168, explored asymptotic behavior of the cardinality of such permutations: $$f_d(n):=\#\{\pi\in\...
T. Amdeberhan's user avatar
5 votes
1 answer
157 views

Furstenberg decomposition for non-compact spaces

Given a topological group $G$, a $G$-space is a topological space $X$ equipped with an action of $G$, such that the map $(g,x) \mapsto g.x$ is continuous. The action is distal if no non-diagonal ...
Colin Reid's user avatar
  • 4,678
3 votes
0 answers
207 views

Compact subgroups of general linear groups over affinoid algebras

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $(A,A^0)$ be a $k$-affinoid algebra, where $A^0$ is the subring of power bounded elements. Suppose given a compact subgroup, L, of $GL_n(A)$, is ...
user105552's user avatar
6 votes
2 answers
678 views

Non-trivial problems about the trivial group

Is there any non-trivial problem (maybe open problem) about the trivial group? I asked already a question about the Laws characterizing the trivial group. There is a description of such laws. As ...
Sh.M1972's user avatar
  • 2,183
2 votes
1 answer
151 views

An atomic solvable Hausdorff topological group with a cardinality greater than that of real line

Is there a Hausdorff topological group $(G,\cal T)$ such that $G$ is a solvable group with a cardinality strictly greater than $\frak c$ and such that there is not any nontrivial (not necessarily ...
Minimus Heximus's user avatar
5 votes
1 answer
168 views

The stabilizer of the conditionally convergent series

The standard rearrangement theorem for conditionally convergent series says that the terms in a conditionally convergent series can be rearranged so that the new sum is any desired number, or $\pm\...
Jeff Strom's user avatar
  • 12.5k
1 vote
1 answer
204 views

What happens when you internalize outer automorphisms?

Given a finitely presented group $G = (Gen|Rel)$, we have a set of inner automorphisms $\{ \phi_a(x) = axa^{-1} | a \in G\}$. Defining the set of outer automorphisms to be those automorphisms of $G$ ...
Samuel Schlesinger's user avatar
2 votes
0 answers
219 views

Fully residual free groups and surface groups

Let $S$ be a surface group of genus $g > 1$ with presentation $$ S := \langle y_1, \ldots, y_g, z_1, \ldots, z_g : [y_1, z_1] \ldots [y_g,z_g] \rangle $$ Is it true that, given a finite subset $...
Vanya's user avatar
  • 581
6 votes
1 answer
825 views

Extra special p-groups

Let $P$ be an infinite extra special $p$-group for some prime $p$, namely, $Z(P)=P'=\Phi(P)$ and $P/Z(P)$ is infinite elementary abelian. Let $C$ be a Prufer $q$-group for some prime $q\neq p$. ...
W4cc0's user avatar
  • 137
4 votes
1 answer
209 views

Why is the image of $G$ under a $B-$adapted homomorphism normal? (Question from Bruhat Tits paper 1)

I posted this question on MSE earlier, however could not elicit a reply. I am not sure if this belongs here, if not, please flag it. Moreover, I am not sure what tags to put on it, so if this question ...
Vishal Gupta's user avatar
3 votes
1 answer
325 views

Reference for real and complex projective representation of finite group

I'm not a mathematician. I've only learnt about irreducible representation of finite group, symmetric group and simple Lie group. In fact, I don't know projective representation belong to which part ...
346699's user avatar
  • 977
4 votes
1 answer
311 views

Kac's lemma for amenable group actions

The classical Kac's lemma says the following. Let $(X,\mu)$ be a probability space and $T$ a measure preserving transformation. Assume $A\subset X$ has positive measure. Then $$\sum_{k\ge 1} k\mu(A_k)...
apvelozo's user avatar
2 votes
1 answer
364 views

New class of finite groups?

I need to consider finite groups $G$ such that for any square-free number $d$ dividing the order of the group $G$, there exists a normal subgroup $H$ in $G$ such that either $H$ or $G/H$ has order $d$...
Taras Banakh's user avatar
  • 40.8k
3 votes
1 answer
339 views

Fourier transform of subgroups of $(\mathbf{Z}/n\mathbf{Z})^*$

Let $\zeta$ be a primitive $n$-th root of unity, and for each function $f : \mathbf{Z}/n\mathbf{Z} \to \mathbf{C}$ define its Fourier transform $\widehat{f} : \mathbf{Z}/n\mathbf{Z} \to \mathbf{C}$ by ...
sercej's user avatar
  • 51
7 votes
0 answers
359 views

When is the character group scheme of a group scheme representable? (Affine Case)

While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...
J. David Taylor's user avatar
11 votes
1 answer
609 views

Virtually free, torsion-free, and locally free groups

A well-known theorem of Stallings says that any finitely generated virtually free torsion-free group is free. Is this true without `finitely generated' condition? In other words, is every ...
Anton Klyachko's user avatar
25 votes
0 answers
960 views

Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group

Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$? I have no real ...
David E Speyer's user avatar
1 vote
0 answers
179 views

Cocompact (finite covolume) lattices in euclidean groups

1) Is there a classification of cocompact ( or finite co-volume) lattices in Euclidean groups E(n)( motions of Euclidean space) ( especially in dimensions 2,3,4)? 2) Also what is (if any) the ...
JasonK's user avatar
  • 11
1 vote
0 answers
354 views

Fuchsian groups and surface groups

The following question may be trivial or inappropriate; I am not sure though. It is known that a cocompact oriented Fuchsian group $\Gamma$ admits a presentation: for given $m,g,d_i \geq 0$ $$ \...
Vanya's user avatar
  • 581
3 votes
0 answers
210 views

Smallest number $n$ for which we don't know the classification of all groups of order $n$

I noticed that in groupprops and wikipedia there are often given tables of classifications of groups of small order. This motivated me to ask, what is the current state of research in classifying all ...
Sidharth Ghoshal's user avatar
8 votes
0 answers
178 views

Zappa-Szép products of the group of integers with itself

Since my previous question didn't get much attention and I couldn't make any relevant progress on it, I thought it would be a good idea to "simplify" it by replacing monoids by groups. That is: ...
HeinrichD's user avatar
  • 5,402
9 votes
0 answers
471 views

"A remarkable Moufang loop"

The 1985 paper A simple construction of the Fischer-Griess monster group by Conway refers to an "in press" article, A remarkable Moufang loop, with an application to the Fischer group $Fi_{24}$, by ...
David Roberts's user avatar
  • 33.8k
9 votes
1 answer
384 views

Groups determined by their group ring and direct products

In the paper [W. Kimmerle - R. Lyons - R. Sandling - D.N. Teague: Composition factors from the group ring and Artin's theorem on orders of simple groups, Proc. London Math. Soc. (3) 60 (1990), no. 1, ...
Amir Baghban's user avatar
2 votes
1 answer
200 views

Endomorphism of the symmetric group of the set of positive integers via action on the prime numbers

For a positive integer $n$, let $p_n$ denote the $n$-th prime number. Further let $f: {\rm Sym}(\mathbb{N}) \rightarrow {\rm Sym}(\mathbb{N})$ be the monomorphism which maps a permutation $\sigma$ to ...
Stefan Kohl's user avatar
  • 19.5k
3 votes
1 answer
281 views

About normalizers of infinite cyclic subgroups of Hilbert modular group

Consider $k$ a totally real finite extension of degree $n$ of $\mathbb{Q}$, i.e., all embeddings of $k$ in $\mathbb{C}$ have their image contained in the field of reals. Denote by $\mathcal{O}_k$ the ...
Luis's user avatar
  • 31
2 votes
1 answer
256 views

Countable union of non Zariski-dense homomorphisms

Let $F_k$ be a free group in $k>1$ letters, and $G$ a semi-simple algebraic group defined over reals $\mathbb{R}$. Consider the representation variety Hom$(F_k,G(\mathbb{R}))$. The points of this ...
Vanya's user avatar
  • 581

1
68 69
70
71 72
159