All Questions
Tagged with reference-request gr.group-theory
700 questions
10
votes
1
answer
207
views
Nonsolvable finite quotients of matrix groups
Suppose that $\Gamma$ is a finitely generated nonsolvable subgroup of $GL(n, R)$. Is it in the literature that $\Gamma$ has a nonsolvable finite quotient? I know how to prove it (the hardest ...
- 31.2k
5
votes
1
answer
329
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 prosolvable,...
- 11.3k
5
votes
2
answers
439
views
Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$
If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$.
When is it ...
- 178
5
votes
1
answer
472
views
Countable reduced abelian group containing all countable reduced abelian groups
Is there a countable abelian group for which its subgroups are exactly all of the countable "reduced" abelian groups? (Reduced means that its divisible subgroup is zero)
Is the following group a ...
- 165
2
votes
1
answer
210
views
Reference proving $\beta^{(2)}_1(G) \le d(G)-1$
I saw a paper that said it is well-known that for finitely generated group $G$: $\beta^{(2)}_1(G) \le d(G)-1$, but I can't find any reference proving it.
$d(G)$ denotes the minimal number of ...
- 197
2
votes
0
answers
1k
views
Lifting of group homomorphisms
I asked this question a few days ago on math stackexchange but didn't get any answer so I thought I post it here too (see here):
In my first course on algebraic topology I heard about the following:
...
- 721
8
votes
1
answer
633
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 ...
- 356
7
votes
2
answers
780
views
Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 52.9k
6
votes
4
answers
596
views
Is the conjugacy problem solvable in $Out(F_n)$?
There is a paper of Martin Lustig on his webpage giving a positive answer to the conjugacy problem for the outer automorphism group of the free group $F_n$. On the other hand, there seems not to be a ...
- 12.1k
3
votes
2
answers
337
views
Frobenius Groups of Automorphisms
Recently, I am looking different papers on the topic
$$\mbox{Frobenius groups of automorphisms of a group.}$$
But I am familiar with Frobenius groups only; not with their (faithful) actions on groups. ...
- 261
13
votes
4
answers
1k
views
Simple groups with the same cardinality as PSL_2(Z/p)
In an undergrad honors algebra course it's sometimes shown that when $p$ is prime and $>3$ then
$PSL_2(Z/p)$ is simple of of order $p(p-1)(p+1)/2$. But that this is the "only" simple group
having ...
0
votes
1
answer
434
views
Reference request: Any connected Lie group has a countable base for its topology
I am looking for a reference for the assertion in the title. This assertion is proved in a comment of user nfdc23 to this question. Has any proof of this assertion been published?
- 14.2k
5
votes
1
answer
458
views
Lower Central Series of Pure Braid Groups?
What is the lower central series $\Gamma_k(P_n)$, where $P_n$ is the pure braid group with $n$ strands? We know that $P_n$ is generated by elements $A_{i,j}$; do we know the generators of $\Gamma_k(...
- 1,108
2
votes
0
answers
417
views
An equivariant Hahn embedding theorem?
The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...
- 1,089
10
votes
1
answer
2k
views
Unitary representations of the ax+b group: an accessible presentation
The "ax+b group" is the group of affine transformations of $\mathbb R$. It is a locally compact non unimodular group.
Its space of irreducible, continuous unitary representations has been described ...
- 9,486
0
votes
0
answers
140
views
Finite group and cyclic cover
Suppose the finite group $N$ surjects to finite group
$F$. It is true that for any $G = N ⋊_α \mathbb{Z}$ there are infinitely many covers of $G$ that are cyclic and
surject to $F$.
But is this ...
- 1,755
1
vote
1
answer
107
views
Name of the class of linearly ordered groups with no minimal positive element
Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that $e<h<g$?
- 5,409
15
votes
2
answers
870
views
A space of ideals
Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...
- 25k
3
votes
2
answers
469
views
Maximal size of minimal generating set
Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer ...
- 5,450
2
votes
1
answer
514
views
Any representation is a subrepresentation of a direct sum of the regular representation
I need a reference for the following statement:
Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $...
- 661
2
votes
1
answer
255
views
Polya-MacMahon-Burnside's generating function at "-1"
$\mathbb{Z}_n$, as a cyclic subgroup of symmetric group $\mathfrak{S}_n$, acts on $[n] :=\{1, 2,\dots,n\}$. Hence $\Bbb{Z}_n$ permutes the elements of the Boolean algebra $2^{[n]}$ of all subsets of $[...
- 43.2k
1
vote
2
answers
742
views
Automorphism group of the affine groups AGL(n,q), ASL(n,q)
I have a question. The automorphism group of the linear groups $GL(n,q)$, the group of linear transformations of $V = \mathbb{F}_q^n$, and $SL(n,q)$, the subgroup of $GL(n,q)$ consisting of elements ...
- 731
6
votes
2
answers
483
views
Products of elliptic isometries
A well-known property on groups acting on trees is:
Theorem: Let $T$ be a tree and $g,h \in \mathrm{Isom}(T)$ two elliptic isometries. If $\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$ then the ...
- 2,371
3
votes
0
answers
170
views
What is $G_2(2^m)$, and how is it embedded in $\Gamma L_6(2^m)$?
I am trying to understand the classification of doubly transitive groups, specifically the nonsolvable affine case. Dixon and Mortimer (p.244) says there are three infinite families, one of which is $\...
- 7,633
12
votes
2
answers
523
views
A question on some computation of group cohomologies
Let $G=H\times J$, where $H\cong J\cong C_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group ...
- 123
4
votes
1
answer
214
views
A quotient group of a self-normalizing spherical subgroup
Let $G$ be simply connected, simple algebraic group over $\mathbb{C}$.
Let $H\subset G$ be a self-normalizing spherical subgroup of $G$,
not necessarily connected or reductive.
Here "self-...
- 14.2k
4
votes
3
answers
272
views
Results about the existence of solutions in groups
Let $G$ be a group. Consider an arbitrary equation given by $w(\vec{g})=e$, where $w: G^n \to G$ takes an $n$-tuple $(g_1,...,g_n)$ to some expression involving products of the $g_i$, their inverses ...
- 223
5
votes
1
answer
884
views
solvable word problem without algorithm
Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...
- 829
9
votes
2
answers
485
views
Reference for restriction of a simple module over a splitting field to a smaller field?
This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group representations....
- 52.9k
0
votes
1
answer
227
views
Finite group cohomology with roots of unity as coefficients
Let $G$ be a finite group of order $n$, and let $L := (\mathbb{Q}/\mathbb{Z})^d$ be a (not necessarily trivial) $G$-module (we assume that $d$ is finite).
By a direct limit argument, there must be a ...
- 11.3k
17
votes
3
answers
1k
views
How to find more (finite almost simple) groups with a given Sylow subgroup
I'm looking for some examples of actions on Sylow p-groups, and often those actions appear in the case of finite almost simple groups. Given a finite almost simple group, I understand in principle ...
- 10.7k
7
votes
0
answers
229
views
Computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$
Do you have a nice modern reference where I could find the computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$, where the action is trivial ?
I have looked at the very few books on cohomology of groups ...
- 1,766
2
votes
1
answer
197
views
Hall $\pi$ subgroups that controls its own fusion
Theorem:Let $P\in Syl_p(G)$ for a finite group $G$. Then $G$ has a normal $p-$ complement if and only if $P$ controls its own fusion.
I wonder if similar argument is true for Hall subgroups (in ...
- 1,169
2
votes
0
answers
66
views
One-parameter group of nonvanishing vector field
Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.
Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure ...
- 1,915
7
votes
0
answers
1k
views
A "direct" proof that hyperbolic groups are not amenable
I am looking for a proof that a finitely generated hyperbolic groups is non-amenable [unless it is virtually cyclic] which is as "metric/combinatoric" as possible.
Here are the two proofs I am aware ...
- 4,432
9
votes
1
answer
573
views
$SL_2(\mathbf{Z},8\mathbf{Z})$ differs from $E_2(\mathbf{Z},8\mathbf{Z})$. Has this result appeared in the literature?
I know a proof that the congruence subgroup $SL_2(\mathbf{Z},8\mathbf{Z})$ differs from its subgroup $E_2(\mathbf{Z},8\mathbf{Z})$, but can't find this fact in the literature. Does anyone know a ...
- 127
13
votes
0
answers
182
views
What do we call a functor of orbits and isomorphisms?
If $G$ is a finite group, then inside the category of $G$-sets and $G$-maps there is the subcategory whose objects are the orbits (transitive $G$-sets) and whose morphisms are the isomorphisms. I have ...
- 55.9k
5
votes
0
answers
169
views
In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
10
votes
2
answers
687
views
Embedding in f.p. simple groups
Dear All!
At the time when Lyndon and Schupp wrote their book there was an open question:
Question: Does every finitely presented group with soluble word problem embed in a finitely presented simple ...
- 1,437
5
votes
1
answer
906
views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups $\...
- 10.4k
3
votes
4
answers
570
views
A polynomial homomorphism from Gl to the group of units is a power of the determinant
I was browsing MO and stumbled upon this post, and I got very curious. I searched for about half an hour and could not find a proof for the statement that any polynomial group homomorphism $\mathrm{Gl}...
- 4,902
10
votes
1
answer
377
views
Fixed set of order p automorphism of Bruhat-Tits tree
I would like to know the structure of the fixed set of an order $p$ automorphism [Edit: induced by a matrix in $GL_2(K)$] on the Bruhat-Tits tree for a p-adic field $K$, specifically in the case where ...
- 17.4k
0
votes
0
answers
353
views
Inner products on abelian groups and general modules
Where can I find a discussion about inner products on something more general than vector spaces, and to which extent should one attempt such a generalization?
My particular interest is in abelian ...
- 229
7
votes
1
answer
808
views
Reference: Finite $p$-Groups
Hall and Blackburn made important contributions in the study of regular $p$-groups and $p$-groups of maximal class. From their work, one can understand that in the classification of groups of order $p^...
- 71
0
votes
0
answers
328
views
Infinite groups in which every element is a commutator
Finite groups in which every element is a commutator are considered in many works. How about infinite group case? Are there any recent results or constructions related to infinite groups in which ...
- 379
4
votes
1
answer
381
views
Are braid groups conjugacy separable?
I would like to re-ask a question that was raised in the comments here:
Normal subgroups of braid groups
Namely, a group $G$ is called conjugacy separable, if it holds that for every two elements $x,...
- 5,039
3
votes
0
answers
184
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 2,087
1
vote
1
answer
255
views
Reference for the delooping of groups (or rings)
This nlab article defines the delooping groupoid of a group, but it does not provide a reference. I am using deloopings of rings, and I would like to refernce to a more classical source.
Could you ...
- 349
6
votes
3
answers
505
views
Irreducible mod-p representation of a semidirect product with trivial p-core
Consider the group $G=H\rtimes{}C$, where $H$ has order prime with $p$ and $C$ is cyclic of order $p^k$, and $C\rightarrow{}\mathrm{Aut}(H)$ is faithful (or equivalently $G$ has trivial $p$-core). ...
- 1,269
3
votes
1
answer
137
views
Subalgebras with finite codimension
In group theory it is well-known that every subgroup of finite index contains a normal subgroup of finite index. It is not true in general that for Lie algebras every subalgebra of finite codimenslon ...
- 379