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

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3
votes
1answer
179 views

Homology of solvable Lie groups made discrete

In what follows "homology" will mean group homology, i.e. $H_*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology. It is well-known how to compute the homology of abelian groups, ...
6
votes
1answer
145 views

Are the integer matrices in SO(3,2) “boundedly generated”?

Let $G$ be the subgroup of integer matrices in $\mathrm{SO}(3,2)$. (The invertible linear maps from a $5$ dimensional real vector space to itself which leave invariant a nondegenerate symmetric ...
1
vote
0answers
60 views

Salvaging Howson's theorem for free profinite groups

This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem? Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...
3
votes
2answers
230 views

Do limit groups satisfy Howson's theorem?

Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must $A \cap B$ be finitely generated? Recall that a limit ...
1
vote
2answers
194 views

Is $K\cap \langle H\cup N\rangle‎\subseteq‎ \langle H\cup (K\cap N)\rangle$? [closed]

If $G$ is a arbitrary group, $H,K,N\leq G$ such that $H‎\subseteq‎ K$, then $K\cap \langle H\cup N\rangle‎\subseteq‎ \langle H\cup (K\cap N)\rangle$?. If it is not true, how can I find an ...
9
votes
3answers
320 views

Integer matrix that does not belong to a free group of rank 2

I'm given two matrices in $SL_2(\mathbb{Z})$ $$ A = \left(\begin{array}{cc} 2 & 3\\ 3 & 5 \end{array}\right), \ \ B = \left(\begin{array}{cc} 5 & 3\\ 3 ...
1
vote
0answers
135 views

Invariant free factors for automorphisms of free products

I am thinking the following question: for a group that splits as a free product $G = G_1 \ast F_n $, where $G_1$ is any infinite group (preferebly freely indecomposable) and we denote by $F_n$ the ...
0
votes
1answer
122 views

Averaging measurable functions over amenable group actions

Let $G$ be an amenable group acting on a space $X$. Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$. Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...
1
vote
0answers
129 views

Subgroup generated by elements of finite order

Let G be any group and let H be the subgroup of G generated by all elements of finite order. Is there a name for H?
10
votes
2answers
292 views

Conjugacy of matrices of order three in $PGL(2,k)$, where $k$ is any field

We know that every matrix of order three in $PGL(2,\mathbb Z)$ is conjugate to the following matrix $$ \left( \begin{array}[cc] &1 & -1 \\ 1&0 \end{array} \right) $$ I want to know if ...
0
votes
1answer
198 views

Computational Algebra and Symbolic Computation - Where? [closed]

Following the line of this question, I'm in my last year of M.Sc., and I'm looking for a place where I can start my PHD. Since that question has been asked 4 years ago, I thought it may be wise to ask ...
4
votes
1answer
169 views

Existence of some character degree in a solvable group

Let $p=2^n-1$ be a Mersenne prime. We know that $H=PSL(2,p^4)$ has an irreducible character of degree $ p^4$. Is there any solvable group of order $H$ with an irreducible character of degree $ p^4$. ...
8
votes
1answer
129 views

Embedding abelian cancellative Hausdorff topological semigroups into abelian Hausdorff topological groups

An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a ...
0
votes
0answers
117 views

Rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$

What is the rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$, where $n\geq 3$ ? A reference would be appreciated. For the definition of "cohomological dimension of a group ...
4
votes
1answer
186 views

reference on classfication of multiply transitive permutation groups

It seems to be true that multiply transitive permutation groups have been classified completely (using CFSG), but I am having trouble finding a reference where this classification is actually stated. ...
3
votes
1answer
96 views

Unipotent radical of minimal parabolic subgroup of a unitary group over an arbitrary field

I am looking for an explicit description of the unipotent radical of a minimal parabolic subgroup of a unitary group, i.e. the group of isometries of a hermitian form, over an arbitrary field. In his ...
0
votes
0answers
61 views

Homology of product of two groups [duplicate]

There is well known formula for the homology of product of two groups with coefficient in integers, that is $0 \rightarrow \oplus_{p+q=n}H_p(G,\mathbb{Z}) \otimes H_q(H,\mathbb{Z}) \rightarrow H_n(G ...
2
votes
0answers
72 views

On subdirect products [closed]

I'm sorry if this question doesn't fit with MO rules, but I've asked on Math SE yet without answers, so I post here with the hope someone will answer me. I want to show, knowing the Goursat's ...
5
votes
0answers
254 views

Quotients of an extension of the Higman group

(Note: this started as a different question that soon changed form, thanks to the answers.) Let $G = \mathbb{Z}/4\mathbb{Z} \ltimes H_4$, where $H_4$ is the Higman group and $\mathbb{Z}/4\mathbb{Z}$ ...
2
votes
0answers
71 views

Positive definiteness via Folner sequence

Let $\Gamma$ be a countable discrete group. A function $\phi:\Gamma\to\mathbb{C}$ is called positively definite if for every finite subset $F$ of $\Gamma$, the |F| by |F| matrix ...
4
votes
2answers
318 views

Structure of the group generated by two specific symplectic matrices

Consider the following two symplectic matrices $$ A \ = \ \left(\begin{array}{rrrr}% 1&0&0&0\\% 0&1&0&0\\% 0&0&-1&1\\% 0&0&-1&0\\% \end{array}\right), \ ...
4
votes
1answer
247 views

Recognize this countably generated abelian group?

I recently came across a construction that, in abstraction, leads to the following family of abelian groups: Fix $1<q<p$ with $q$ and $p$ relatively prime. The group $G_{(p,q)}$ is given by the ...
10
votes
1answer
353 views

Extending an infinite simple group

Maybe the question does not fit here. Yesterday in my logic course, I presented a nice example about an application of model theory to group theory. The example is due to Hodges and as following: For ...
6
votes
2answers
273 views

Faithful projective representations of symmetric groups

This is a reference request. Do you know where I can find the dimensions of the faithful projective representations of $S_n$ and $A_n$ for $n\ge 5$? Thank you in advance.
7
votes
3answers
525 views

Beyond Brauer's theorem

Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary ...
17
votes
1answer
633 views

“Squeezing” a finite group between symmetric groups

For a finite group $G$, take $m$ as the largest integer such that $G$ has a subgroup $H\cong S_m$ and $n$ as the smallest integer such that $G$ is itself isomorphic to a subgroup of $S_n$. We then ...
1
vote
0answers
133 views

Is there a brute force method for determining irreducible representations?

Suppose I have some groups $G_1$, $G_2$, $G_3$, etc... Then the direct product is given by $G = G_1 \times G_2 \times G_3 \ldots$ I know that the sub-representations of a reducible representation ...
-1
votes
1answer
115 views

Example of a group in which centralizers of every element are non-abelian [closed]

I am studying $AC$-groups, i.e. groups in which the centralizer of every non-identity element is abelian. Now I need an example of a group in which the centralizer of every non-identity element is ...
2
votes
0answers
33 views

Local zetafunction of T-group and Lie-ring coincide

Grunewald, Segal and Smith define in "Subgroups of finite index in nilpotent groups" a zetafunction associated to a finitely generated, torsion-free nilpotent group G by counting normal subgroups ...
11
votes
1answer
226 views

A generalization of residual finiteness to topological groups

Consider the following generalization of residual finiteness to topological groups. A locally compact Hausdorff group $G$ is called residually compact if for every compact $K \subseteq G$ there is a ...
2
votes
0answers
109 views

Divisible fundamental group [duplicate]

I apologize if this question seems trivial or elementary. Is there any concrete topological space with divisible fundamental group? For example, is there any such a space the fundamental group in ...
4
votes
1answer
264 views

Number of different positions of rooks on chessboard

I know that this topic as been mentioned before, but no accurate answer has been provided. Suppose we have to place $n$ rooks on $n \times n$ chessboard so that no one attacks another. How to count ...
1
vote
0answers
95 views

A finite distributive lattice which may be represented as the normal subgroup lattice of a supersolvable group

Is there a supersolvable group $G$ with the lattice of all its normal subgroups, order-isommorphic to the 18-element lattice of down-sets of this poset: ? It has been proved that not every finite ...
5
votes
0answers
108 views

Connectedness of cones in the boundary of a 1-ended hyperbolic group

Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...
2
votes
1answer
92 views

Looking for a modern source about Ulm Invariants

I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm ...
4
votes
1answer
147 views

Obstruction for two subgroups to be conjugated by an automorphism

Altough this sounds as a very basic question, I didn't receive any answer on stack exchange and by people more knowledgeable than me Take $p$ a prime number and $P$ an abelian finite $p$-group. Let ...
13
votes
4answers
535 views

Normal subgroups of an extension of the Higman group

Let $G = \mathbb{Z}/4\mathbb{Z} \ltimes H_4$, where $H_4$ is the Higman group and $\mathbb{Z}/4\mathbb{Z}$ acts on $H_4$ in the obvious way (permuting the four standard generators cyclicly). The group ...
5
votes
0answers
117 views

A group acting acylindrically on a fine hyperbolic graph with infinite edge stabilizers

I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that a) the graph $\Gamma$ is fine, b) $\Gamma$ is not a tree, c) not all ...
12
votes
1answer
420 views

Realizing symmetric groups by diffeomorphisms

Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x_1, \dots, x_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x_i$ surjects ...
4
votes
0answers
94 views

Example of “exotic” verbal subgroups of free groups

This will be an ambiguous question. I am interested in various examples that appear in the literature of verbal subgroups of free groups, but which are not part of the "classical examples" like ...
6
votes
2answers
379 views

Counting cyclic subgroups of order $p^{2}$: $p$ an odd prime vs. $p=2$

While playing with Frobenius' problem (about finite groups $G$ in which, for some positive integer $n \mid |G|$, there are exactly $n$ elements of order dividing $n$), I came up with the following ...
3
votes
1answer
157 views

Intersection of maximal subgroups of PSL(2,q)

Let $G := PSL(2,2^r)$, and let $M$ be a maximal subgroup of $G$ isomorphic to $PSL(2,2^s)$. I need to compute $H := M \cap M^g$ for $g \in G-M$. It seems to me that $|H|$ must be $2^r, 2^r\pm 1$ or ...
6
votes
2answers
360 views

Counting matrices over finite fields of a given order

How can I count/enumerate matrices in ${\rm GL}(2,{\rm GF}(2^5))$ of order $3$? In general, how can I obtain the number of matrices in ${\rm GL}(2,{\rm GF}(q))$, where $q$ is a power of a prime, of ...
5
votes
0answers
41 views

“Quasi-orthogonal” subgroup of a group with length?

For my project in bivariant K-theory for locally convex algebras, I'm looking how to call a particular notion of groups, too simple to be never considered elsewhere. Let $G$ b a group with length ...
8
votes
2answers
268 views

Groups with trivial rational homology and their finite index subgroups

For a short exact sequence $0 \to G \to H \to K \to 0$ of (discrete) groups with $K$ finite we have, as a consequence of the Hochschild-Serre spectral sequence, that $H^{\ast}(H;\mathbb Q) = ...
7
votes
2answers
304 views

How to compute second homology of a group given by presentation with two relators

I am interested in calculating of $H_2(G,\mathbb{Z})$, where $G$ is a group given by presentation with two relators $\langle a,b| r_1 = r_2 = 1\rangle$. Moreover, I am interested in such ...
7
votes
2answers
229 views

What is the most efficient way to factor a matrix into a given set of generators?

I am studying finite index subgroups of certain finitely presented groups. The particular conditions on my groups make this problem easier than I phrase it here, but I am curious about a more general ...
12
votes
4answers
908 views

Number of squares in a finite group

This was asked at MSE but never answered. Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if ...
2
votes
0answers
219 views

Properties of Higman's group

The infinite group of Higman which has no finite quotient is given by the presentation (with 4 generators and 4 relations): $$ G = \langle a_i, i \in \mathbb{Z}/4\mathbb{Z} \mid a_ia_{i+1 \,(\text{mod ...
6
votes
1answer
217 views

Graph with group structure?

Is there any established theory of graphs which themselves are groups? I don't mean Cayley graphs or "graphs of groups". I mean a graph whose set of vertices forms a group, where the group operation ...