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

**3**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**2**answers

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

**3**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**4**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**2**answers

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

**4**answers

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

**0**answers

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

**1**answer

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 ...