Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,922
questions
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(\...
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 ...
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{...
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. $$w(G) = \bigcap_{H \triangleleft \triangleleft ...
2
votes
0
answers
256
views
Is conjugacy problem hard in braid group?
Recently I studied the braid group and conjugacy problem. It is believed that conjugacy problem is hard on braid group. My friend gave me an EXE file, and I use it for solving conjugacy problem, as an ...
7
votes
1
answer
710
views
Characterization of group characters
I wonder how to characterize the characters of a (say, finite) group $G$ as special class functions, in particular for the case $G=S_n$ (symmetric group). The answer to this is presumably well known ...
28
votes
0
answers
657
views
Mathieu group $M_{23}$ as an algebraic group via additive polynomials
An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
2
votes
1
answer
143
views
Right socle of a group ring
Let $p$ be a prime number and $n$ a positive integer. I want to know what is the (right) socle of the group ring $A=\mathbb Z_{(p)}C_n$, where $\mathbb Z_{(p)}$ is the localization of integers at the ...
6
votes
1
answer
372
views
One-ended finitely presented subgroups of hyperbolic groups
In Hyperbolic groups (page 82), Gromov claims that, by a standard application of Thurston's method of geodesic (hyperbolic) simplices, it can be prove that a hyperbolic group contains finitely many ...
4
votes
2
answers
444
views
Number of torsion-free abelian groups
Let $\mathfrak{c}$ be the cardinality of the continuum. How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of ...
2
votes
1
answer
183
views
Number of homomorphism, or number of solution to equations, in finite groups
Let $G$ be a finite group, and let $P$ be a finitely generated group.
Consider the number $$n=\#Hom_{Grp}(P,G).$$
It is known (see Number of solutions to equations in finite groups) that under ...
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 ...
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 ...
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$, ...
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 ...
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?
...
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_{...
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 ...
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 ...
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 ...
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?
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 ...
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$ ...
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: $\...
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\...
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 ...
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 ...
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 ...
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 ...
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\...
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 $G$ ...
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 $...
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 $q\neq p$.
...
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 ...
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 ...
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 $$\sum_{k\ge 1} k\mu(A_k)...
2
votes
1
answer
364
views
New class of finite groups?
I need to consider finite groups $G$ such that for any square-free number $d$ dividing the order of the group $G$, there exists a normal subgroup $H$ in $G$ such that either $H$ or $G/H$ has order $d$...
3
votes
1
answer
339
views
Fourier transform of subgroups of $(\mathbf{Z}/n\mathbf{Z})^*$
Let $\zeta$ be a primitive $n$-th root of unity, and for each function $f : \mathbf{Z}/n\mathbf{Z} \to \mathbf{C}$ define its Fourier transform $\widehat{f} : \mathbf{Z}/n\mathbf{Z} \to \mathbf{C}$ by
...
7
votes
0
answers
359
views
When is the character group scheme of a group scheme representable? (Affine Case)
While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...
11
votes
1
answer
609
views
Virtually free, torsion-free, and locally free groups
A well-known theorem of Stallings says that
any finitely generated virtually free torsion-free group is free.
Is this true without `finitely generated' condition?
In other words,
is every ...
25
votes
0
answers
960
views
Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group
Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$?
I have no real ...
1
vote
0
answers
179
views
Cocompact (finite covolume) lattices in euclidean groups
1) Is there a classification of cocompact ( or finite co-volume) lattices in Euclidean groups E(n)( motions of Euclidean space) ( especially in dimensions 2,3,4)?
2) Also what is (if any) the ...
1
vote
0
answers
354
views
Fuchsian groups and surface groups
The following question may be trivial or inappropriate; I am not sure though.
It is known that a cocompact oriented Fuchsian group $\Gamma$ admits a presentation: for given $m,g,d_i \geq 0$
$$
\...
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 ...
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:
...
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 ...
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, ...
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 ...
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 ...
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 ...