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

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vote

**1**answer

85 views

### embedding of finite groups into product

Our situation is following. Assume that we have free product $\star_{i<n} G_i$ each $G_i$ finite group and assume that we have normal subgroup $K$ such that composition of canonical embedding and ...

**21**

votes

**2**answers

767 views

### Groups where word problem is solvable, but not quickly?

Are there finitely generated groups whose word problem is solvable, but not quickly? It would be great to have specific examples, but existence results would also be helpful.
All of the groups that ...

**6**

votes

**1**answer

176 views

### GIT quotients and automorphisms

Let $X$ be a smooth projective variety. Then we have an exact sequence:
$$0\mapsto Aut^{o}(X)\rightarrow Aut(X)\rightarrow H\mapsto 0$$
where $Aut^{o}(X)$ and $H$ are respectively the connected ...

**1**

vote

**0**answers

107 views

### Classification of Automorphism set of a Regular graph

Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation ...

**2**

votes

**1**answer

110 views

### Projectivity of torsion-free modules over integral group rings

Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.
If we assume ...

**5**

votes

**2**answers

416 views

### What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?

Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$:
\begin{eqnarray*}
h: (x, y, z) &\mapsto& (x, y, xy - z) \\
u: (x, y, z) ...

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votes

**0**answers

59 views

### Is there a general theory for the structure of the (semi)group generated by morphisms of an affine space $F_p^3$?

Consider an affine space $\mathbb{F}_p^3$, and assume we have a handful of morphisms $f_i : \mathbb{F}_p^3 \rightarrow \mathbb{F}_p^3$ given by $$f_i(x, y, z) =(P_i(x, y, z), Q_i(x, y, z), R_i(x, y, ...

**3**

votes

**1**answer

109 views

### Commutator subgroups as normal supplmements

The following question has been asked about a week ago on MathUnderflow (no answers).
Let $F$ be a free group and let $N$ be a normal subgroup of $F$ such that
\begin{equation*} \tag{*}
F = [F,F] ...

**1**

vote

**1**answer

220 views

### Generalization of a theorem of Steinberg

Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type:
...

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votes

**2**answers

388 views

### C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$

In 1977, Joan Plastiras gave a striking example of two non $*$-isomorphic C$^*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M_2(\mathbb C) \simeq \mathcal B\otimes ...

**8**

votes

**1**answer

586 views

### Which finite groups have no irreducible representations other than characters?

A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a ...

**-4**

votes

**1**answer

82 views

### Group theory: Find an element from some group G such that order of a is 6 and c(a)=/= c(a^3), where c(a) := {xEG : xa=ax} [closed]

Find an element from some group G such that order of a is 6 and c(a)=/= c(a^3), where c(a) := {xEG : xa=ax}.

**1**

vote

**1**answer

119 views

### Involutive automorphisms of a finite abelian p-group

First, let $A$ be a finitely generated free abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A_i$, where ...

**7**

votes

**0**answers

132 views

### An amenable group containing a wreath product of itself

Does there exist a finitely generated amenable group $G$ which contains a subgroup isomorphic to $G\wr\mathbb{Z} = \bigoplus_{n\in\mathbb{Z}} G \rtimes \mathbb{Z}$?

**2**

votes

**2**answers

237 views

### Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)

In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.
Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff ...

**6**

votes

**4**answers

438 views

### SO$(4)$ (& SO$(n)$) characterization?

I believe it is the case that
any finite subgroup of SO$(3)$
(the $3 \times 3$ orthogonal matrices of determinant $1$)
is either a cyclic group $C_n$,
or a dihedral group $D_n$, or one of the groups ...

**6**

votes

**1**answer

134 views

### presentations of subalgebras

Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x_1,...,x_n$ subject to finitely many ...

**2**

votes

**1**answer

122 views

### Counting elements with certain word length in abelian groups

Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...

**3**

votes

**1**answer

147 views

### Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?

Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$?
My intention ...

**6**

votes

**0**answers

156 views

### Automorphism groups for free groups with action

Let $A$ be a (finitely generated) free group and $G$ be a (finitely generated) free $A$-group - that is, a group with an action of $A$, which is free in the category of groups with an $A$-action. ...

**2**

votes

**2**answers

125 views

### What are the 2-generated subgroups of the special linear group $SL(2, q)$ over a finite field?

What is the subgroup structure of the subgroups $\langle a, b\rangle$ where $a, b \in SL(2, q)$?

**6**

votes

**1**answer

224 views

### Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation
$$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$
In other words, for every finite simple nonabelian group $G$, do there exist ...

**4**

votes

**1**answer

488 views

### Is there a structure theorem or group law for finite groups generated by two elements?

Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...

**1**

vote

**0**answers

84 views

### Space of polynomially growing harmonic functions on a Lie group

There is a recent theorem by Kleiner (based on previous work by Colding & Minicozzi), that reads:
If $G$ is a finitely generated group of polynomial growth, then for every $d$ the space of ...

**0**

votes

**1**answer

210 views

### A question on permutations

Given integers $1$ through $n$, let $S$ be set of ordering of integers that respect even alternating or reverse alternating permutations (https://en.wikipedia.org/wiki/Alternating_permutation) up to ...

**1**

vote

**0**answers

154 views

### divisible 2nd cohomolgy group $H^2(G,\mathbb{Z}G)$

Recall that a (nontrivial) abelian group $A$ is called divisible if the multiplication by $n$ is surjection $A\to A$ for all $n\in\mathbb{Z}_{>1}$.
My question is the following.
Is there a ...

**11**

votes

**2**answers

383 views

### Automorphisms of finite order in $Out(\widehat{F_2})$

Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime.
Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...

**7**

votes

**1**answer

360 views

### What is the normal closure of $GL_2(\mathbb{Z})$ inside $GL_2(\mathbb{Z}_\ell)$?

This weird problem popped up in my research:
Let $\ell$ be a prime. Is there a description of the smallest normal subgroup of $GL_2(\mathbb{Z}_\ell)$ containing $GL_2(\mathbb{Z})$?
Is there a ...

**0**

votes

**0**answers

84 views

### Poincare inequality for connected Lie groups

Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:
symmetric,
adapted (in the sense that there is no proper subgroup $H$ such ...

**2**

votes

**1**answer

104 views

### Inducing representations from the stabilizer of a partition

For each positive integer $i$, let $A_i$ be a fixed representation of the symmetric group $S_i$. I won't tell you exactly what $A_i$ is, but let's say that I have a very explicit description of its ...

**3**

votes

**0**answers

62 views

### Is the family of probabilities generated by a random walk on a finitely generated amenable group asymptotically invariant?

Is the family of probabilities $\mu^n$ (convolution) generated by a random walk $\mu$ on a finitely generated amenable group $G$ asymptotically invariant ($\|g\mu^n-\mu\|_{L^1}\to 0$ for any $g\in ...

**4**

votes

**2**answers

210 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 ...

**2**

votes

**1**answer

175 views

### The first Betti number of a finite covering space of a closed 3-manifold

Let $\tilde{M}\to M$ be a finite covering of closed 3-manifolds.
Is it possible that $\beta_1(\tilde{M})-1> [\pi_1(M):\pi_1(\tilde{M})](\beta_1(M)-1)>0$?
Here, ...

**6**

votes

**0**answers

105 views

### Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks

I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...

**3**

votes

**3**answers

112 views

### Are there references for the properties of words formed in finite groups using L-systems? (In particular, the algae L-system.)

Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} ...

**4**

votes

**0**answers

186 views

### A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a ...

**5**

votes

**0**answers

235 views

### Proving that a subgroup is normal

This question is partly inspired by the recent question on measurability and the axiom of choice.
Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary ...

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votes

**0**answers

101 views

### Cycles covering the edges of the graph corresponding to the Van Kampen diagram of a presentation of a group

Let the group $G$ have the presentation $\langle x_1, \dots, x_n \;|\; r_1, \dots, r_m \rangle$. Let $\Gamma$ be a labelled directed graph corresponding to Van Kampen diagram over the above ...

**2**

votes

**1**answer

82 views

### Just-not-nilpotent-by-compact quotient of a locally compact group

It is known (and not complicated to prove) that for a finitely generated not virtually nilpotent group $G$, we can pass to a quotient $G/N$ of $G$, such that the quotient is just not virtually ...

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votes

**0**answers

142 views

### Cayley graphs with special subgraphs and some related problems

I asked some questions about finite Cayley graphs with special type of subgraphs which has been answered by Dear Prof. Godsil. It can be seen in the MO page with address:
Cayley graphs and its ...

**3**

votes

**0**answers

158 views

### Uniform sub-linearity of sub-additive functions on groups

Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...

**3**

votes

**1**answer

295 views

### Are there enough additive permutations?

I am hoping to learn enough about additive permutations to help with a number theory problem. These permutations also have connections to difference sets, orthomorphisms, transversals, and other ...

**5**

votes

**1**answer

287 views

### Which subgroup order of the symmetric group is the most frequent?

Question: What is the most frequent order of subgroups of $S_n$?
More precisely: Let $a_k$ be the number of subgroups of $S_n$ with order $k$. What is the maximum of $a_k$?
This question came up ...

**2**

votes

**2**answers

362 views

### Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

This is a re-post from MSE (because I did not get the kind of answer I wanted even after offering a bounty).
At the outset I must mention that I don't have a fairly working knowledge of Galois Theory ...

**49**

votes

**1**answer

3k views

### Why can't a nonabelian group be 75% abelian?

This question asks for intuition, not a proof.
An earlier question,
Measures of non-abelian-ness
was thoroughly answered by Arturo Magidin.
A paper by Gustafson1
proves that, for a nonabelian group,
...

**11**

votes

**3**answers

680 views

### Your favorite papers on geometric group theory

I would like to improve my culture on geometric group theory, so I am interested in short and accessible papers on subjects related to this field but which are not always available in the classical ...

**0**

votes

**0**answers

49 views

### Antichains in subgroups up to group automorphism

A family of subgroups of a group G will denote a nonempty collection of subgroups closed under conjugation and further passing to subgroups.
In a Noetherian group (any ascending chain of subgroups ...

**0**

votes

**1**answer

111 views

### Spliting of short exact exact sequences of partially ordered groups

Consider a short exact sequence of partially ordered groups
$$0 \longrightarrow H \stackrel{\alpha}{\longrightarrow} G \stackrel{\beta} {\longrightarrow} G/H \longrightarrow 0 ,$$ where $H$ is a ...

**0**

votes

**1**answer

83 views

### SO(3) transformation that produces a reflection [closed]

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$
$v^T\cdot w=0$,
and the Householder transformation
...

**2**

votes

**0**answers

197 views

### Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy?
...