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

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0
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33 views

Is there a bigger group?

Assume $A,B,C$ are abelian groups, $A$ is a subgroup of $B$ and $A$ is a subgroup of $C$. The question is how to construct an abelian group $D$ satisfies: $A$ is a subgroup of $D$ and there are ...
5
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0answers
37 views

Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...
-1
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1answer
130 views

Why do we not lose any generality by proving it only for finitely generated groups

In the proof of following theorem, in a paper by Farkas- Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring ...
4
votes
1answer
87 views

Kleinian groups containing an isomorphic copy of itself

Is there any example of a Kleinian group (acting on $\mathbb{H}^n$, $n \ge 3$) that contains a finite index isomorphic copy of itself? Here I don't consider Kleinian groups that only have parabolic ...
19
votes
3answers
555 views

Is every abelian group a colimit of copies of Z?

More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$? Note that this does not follow ...
0
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0answers
30 views

Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have minimum feedback vertex set of size $\Omega(n)$?

Feedback vertex set is a set of vertices whose removal leaves an acyclic graph. It is known that every vertex transitive graph on $n$ vertices has minimum vertex cover of size $\Omega(n)$. It is also ...
5
votes
2answers
195 views

Extensions of $SL(2,\mathbb{F}_q)$

Let $n = q(q^2-1)$ (the order of $SL(2,\mathbb{F}_q)$ I think). How many groups of order $2n$ contain $SL(2,\mathbb{F}_q)$ as a (necessarily normal) subgroup? Is this number known exactly? Seems ...
1
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0answers
56 views

explicit zero 2-cocycle

Let $G$ be a group which acts linearly on a vector space of dimension $n$ over a field $k$. Denote by $\rho$ this representation and consider the associated adjoint representation $Ad\rho$ which is ...
2
votes
3answers
321 views

A table for irreducible integral representation of finite cyclic groups

Is there such a table where the irreducible integral representations of finite cyclic groups are listed? Edited: Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
1
vote
1answer
86 views

Projectors onto the invariant subspaces of a unitary representation $U \otimes U^* \otimes U \otimes U^*$

Let $$U \mapsto U \otimes U^* \otimes U \otimes U^*$$ be a unitary representation of the unitary group $U(n)$ acting on the vector space $V$ (where $U^*$ is the complex conjugate of $U$). We can ...
43
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3answers
1k views

Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...
4
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0answers
70 views

A relation on the set of isomorphism classes of finitely generated groups

Let $G$ be the set of finitely generated groups up to isomorphism hence its elements will be noted $[B]$ where $B$ is some finitely generated group. On this set we put a relation $\mathcal{ND}$ ...
1
vote
1answer
91 views

Collections in direct products and freeness

I am looking for references about the following type of questions: Let $G$ and $H$ be two groups, let $(g_i:i\in I)\subset G$ and $(h_i:i\in I)\subset H$ be collections of group elements, and ...
5
votes
0answers
155 views

What is the kernel of the map $Out(\widehat{F_2})\rightarrow GL_2(\widehat{\mathbb{Z}})$?

Let $F_2$ be the free group on two generators, and $\widehat{F_2}$ its profinite completion. Let $Out(\widehat{F_2})$ be the outer automorphism group of $\widehat{F_2}$, ie, $Out(\widehat{F_2}) = ...
19
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0answers
228 views

Guess that group via product queries

Suppose someone (person B) knows a finite group $G$ of order $n$. You (person A) know only the order $n$, and that $1$ is the name of the identity element. The group elements are named $1,2,\ldots,n$ ...
9
votes
2answers
211 views

Embedding $\mathrm{PGL}(n,q^h)$ in $\mathrm{PGL}(nh,q)$

It is not very hard to see that for each prime power $q$ and natural numbers $n,h$, we have an embedding $$\iota \colon \mathrm{GL}(n,q^h) \hookrightarrow \mathrm{GL}(nh, q),$$ obtained by choosing a ...
11
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2answers
467 views

Obstructions for a group to be the multiplicative group of a field [duplicate]

It is well known that every finite multiplicative subgroup of a field is cyclic. I somehow got interested in a possible reverse implication: Assume we have an abelian group $G$ whose every finite ...
1
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0answers
106 views

How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know, For which $G$ can the ...
2
votes
1answer
192 views

Explicit examples of Dehn presentations of hyperbolic groups

It is well known fact that a (f.g.) group is hyperbolic if and only if it admits a (finite) Dehn presentation. My question concerns something I'm struggling with since the first time I read the proof ...
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0answers
206 views

Analogues of the Monster for central charges different from 24

One way to define the Monster group is to consider a conformal field theory (CFT) corresponding to central charge $c=24$ and look at the automorphism group of its vertex operator algebra. For one of ...
2
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0answers
80 views

Does every group that satisfies the maximal permutizer condition then satisfy the permutizer condition?

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
10
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1answer
376 views

Class field towers

It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of ...
2
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0answers
90 views

Does Kaplansky's Zero Divisor Conjecture hold valid for (torsion-free) residually finite groups?

Kaplansky's Zero Divisor Conjecture states that the group algebra $KG$ has no zero divisor for any field $K$ and any torsion-free group $G$. Does Kaplansky's Zero Divisor Conjecture hold valid ...
1
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1answer
207 views

Limit Group decomposition

I would need a clarification about a statement in the article Limit groups and groups acting freely on $\mathbb{R}^n$-trees by Vincent Guirardel. First recall that a limit group is a finitely ...
1
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0answers
77 views

When can the rank of a submodule be bigger than the rank of the module itself? [migrated]

It is well known that the dimension of a subspace is less than or equal to the dimension of the vector space it is contained in. The same is true e.g. for modules over a principal ring. I am looking ...
0
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0answers
57 views

Action of semidirect products of cyclic groups

Is there anything known about group actions of $C_{p}\rtimes C_{p}^{*}$ on the ring of real polynomials $\mathbb{R}[X_{1},\ldots,X_{n}]$, where $C_{p}$ denotes the cyclic group of order $p$ and $p$ is ...
5
votes
2answers
85 views

“Relative cone types” for groups relative to some collection of subgroups

It is a well known fact that an infinite hyperbolic group contains an element of infinite order (see e.g. Bridson, Haefliger, Metric spaces of non-positive curvature, Prop. 2.22 on p. 458) I am ...
-1
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0answers
30 views

Rank of finite solvable group [migrated]

I am very interested in the following question. Is there a finite solvable group G with the property that rank G - rank G_ab > n for n > 2? Here G_ab denotes the abelianization of G. For all the ...
4
votes
2answers
137 views

Orbits of the maximal compact subgroup on the light cone for $p$-adic groups

It is known that if $Q$ is an indefinite non-degenerate quadratic form on $ \mathbb{R}^n$ with $n \ge 3$, then any maximal compact subgroup $K$ of the orthogonal group $SO(Q)$ acts transitively on the ...
2
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0answers
62 views

Pro-G_p*G_q topology, profinite topology

Let $W=G_p*G_q$, where $G_p$ and $G_q$ are pseudovarieties of all finite $p$-groups and all finite $q$-groups respectively, with $p$ and $q$ fixed prime numbers. The pro-$W$ topology on a group $G$ is ...
2
votes
1answer
171 views

Uncountable cardinals and Prufer $p$-groups

Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...
3
votes
1answer
119 views

amalgamation of locally finite groups

It is well known that in category of groups there are Push-outs so it is possible to realize amalgamation in some kind of most free way. My question is what about category of locally free groups? I ...
1
vote
1answer
128 views

Generating finite groups using subgroups

For finite groups $L \leq K$ we define $d(L,K)$ to be the least $n \in \mathbb{N}$ for which there exist $a_1, \dots, a_n \in K$ such that $\langle L, a_1 \dots, a_n \rangle = K$. Is there some $m ...
1
vote
0answers
59 views

Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...
3
votes
3answers
434 views

The free group of a group and the kernel of a canonical morphism

Let $G$ be a group and $F_G$ the free group on the set $G$. Then there exists a canonical surjective morphism ${\rm can}: F_G \to G \to 1$ constructed as follows: let $(e_x)_{x \in G}$ be a copy as a ...
14
votes
2answers
348 views

Explicit permutation representation of the Thompson sporadic simple group?

The Thompson group Th of order $90745943887872000$ is one of the sporadic simple groups occurring in the classification of finite simple groups. Its maximal subgroups are known (see ...
2
votes
1answer
78 views

Generators of the colored braid group (two colors), reference

I consider the group $B_{n,n}$, the braids, colored in two colors, say all odd strings are black and all even strings are white. It is easy to find a set of generators for $B_{n,n}$: $$ \begin{cases} ...
0
votes
0answers
45 views

projective representation of supergroup

In fact, I am not very clear about what I am asking, but I am looking for a concrete example of supergroup which has non-trivial projective representation(some supergroup similar to usual Lie group ...
6
votes
1answer
209 views

For discrete groups, does the Haagerup property imply the AP of Haagerup-Kraus?

I don't expect to find an explicit counterexample to my question, because any example which was known to have the Haagerup property yet not have AP would have given an exact group without AP, and the ...
4
votes
0answers
63 views

Topological systems of imprimitivity

Let $G$ be a group acting by homeomorphisms on a topological space $X$. $G$ is topologically transitive if every open $G$-invariant subset of $X$ is empty or dense. Here is an attempt to define ...
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0answers
16 views

Is trace of regular representation in Lie group a delta function? [migrated]

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
4
votes
3answers
311 views

Must the powers of some element always grow linearly with respect to a word metric?

Suppose we have a group $G$ which is finitely generated , and let $|\cdot |$ denote some word metric on it. Must there be an element $a\in G$ such that $|a^n|\ge c\cdot n$ for some $c>0$? My ...
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votes
1answer
157 views

The name of a group of order 24 [closed]

I encountered a group $G =\langle(1,3,2,4),(3,5,4,6)\rangle\subseteq S_6$ in my study, but I do not know its name. Let $f=(1,3,2,4)$ and $g=(3,5,4,6)$. We have $g^2=fg^2f$, and thus $\langle ...
1
vote
0answers
90 views

Homology and Burnside ring

If $G$ is a finite groupe, we denote $\mathcal{S}(G)$ the category of finite $G$-sets and $\mbox{I}(G)$ the set of isomorphism classes of it's objects. The Burnside ring of $G$, denoted by ...
2
votes
0answers
127 views

Groups acting on complexes

Let $G$ be a finite group. We define a $G$-simplicial complex $\mathcal{A}(G)$ with set of vertices $G^*:=G-\{e\}$ and the simplices are the abelian subsets of $G^*$. The groupe $G$ act simplicially ...
2
votes
3answers
183 views

Rank of a special linear group over a finite field

What is the rank (minimal number of group generators) of $SL(n,\mathbb{F})$ in the situation when $SL(n,\mathbb{F})$ is not perfect (i.e. when $SL(n,\mathbb{F})$ is different from $SL(2,\mathbb{F}_2)$ ...
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1answer
94 views

direct product of a finite group with an infinite symmetric group [closed]

Cross-posted from MSE: http://math.stackexchange.com/q/1226622/15624. Let $G$ be any finite group, and $S_{\aleph_0}$ the group of all bijections $\mathbb{Z}\rightarrow \mathbb{Z}$. Is $G\times ...
3
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2answers
382 views

Upper bound of |Aut(G)| for a p-group

If G is a p-group which is finitely generated with order p^n then what is the upper bound of |Aut(G)|.
4
votes
1answer
201 views

Coherent subgroups of $F_2 \times F_2$

A group is coherent if its finitely generated subgroups are finitely presented. For instance, $F_2 \times F_2$ is a well-known example of incoherent group. My question is: Is a subgroup of $F_2 ...
7
votes
1answer
276 views

Products of subgroups of a free group

Let $F$ be a free group, and let $A,B \leq F$ be two subgroups such that $AB$ contains a nontrivial normal subgroup of $F$. Must either $A$ or $B$ contain a nontrivial normal subgroup of $F$? What if ...