# Questions tagged [gr.group-theory]

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

5,691
questions

**3**

votes

**3**answers

156 views

### Subgroup generated by a subgroup and a conjugate of it

Let $H\leq G$ be groups, and $a\in G$ so that $\langle H,a\rangle=G$. Does it follows that $\langle H\cup aHa^{-1}\rangle$ is a normal subgroup of $G$?
My hope is that this is true, and my guess is ...

**0**

votes

**0**answers

33 views

### Matrix logarithm for d-dimensional cyclic permutation matrix

I want to find the matrix $\hat{H}_d$ which, when exponentiated, leads to a d-dimensional cyclic permutation transformation matrix.
I have solutions for d=2:
$$
\hat{U}_2 =\left( \begin{matrix}
...

**1**

vote

**0**answers

27 views

### What are the compact Aut(A) of an algebra A(G), G finite, that contains the identity?

If we have an algebra A over a finite group G, then if G is non-abelian we can have a non-trivial set of compact automorphisms of A that map the elements of G onto a set isomorphic to G. It may be ...

**-3**

votes

**1**answer

78 views

### Orbit size of an element [on hold]

Let $H$ be a normal subgroup of $G$ and assume that $G$ is acting over a set $X$. Let $c$ be some element of $X$, is there any relationship among the size of the orbit of $c$ under the action of $H$ ...

**9**

votes

**2**answers

342 views

### Regular subsets of $\text{PSL}(2, q)$

Suppose a group $G$ acts on a set $\Omega$. Call a subset $A \subset G$ regular (or sharply transitive or simply transitive or...) if for every two points $\omega_1, \omega_2 \in \Omega$ there is a ...

**2**

votes

**1**answer

153 views

### What generalizes symmetric polynomials to other finite groups?

Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...

**5**

votes

**0**answers

120 views

### Explicit description of the smallest class of groups, that contains all finite simple groups and is closed under semidirect products

Suppose $\Pi$ is the smallest class of groups satisfying the following conditions:
All finite simple groups lie in $\Pi$
If $G \cong H \rtimes K$ and both $H$ and $K$ are in $\Pi$, then $G$ is also ...

**1**

vote

**1**answer

53 views

### Structure of extensions arising in Lie approximation of connected groups

My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known:
Let $G$ be a connected, locally compact, Hausdorff group, ...

**5**

votes

**1**answer

100 views

### Analog of the Lie Product formula for commutators

Let $X, Y$ be elements of a Lie algebra. Consider the group $G$ generated by (limits of) arbitrary products of the elements
$$ G = \langle{e^{tX},e^{sY}\rangle}$$
for all $t,s$. The Lie product ...

**6**

votes

**0**answers

121 views

### Finite subquotients of R. Thompson's group $F$

Recall R. Thompson's group $F$ acting on the interval $[0,1]$: it consists of piecewise linear oriented maps with slopes a power of $2$ and dyadic breakpoints.
Is every finite subquotient (= quotient ...

**2**

votes

**0**answers

100 views

### On group varieties and numbers

Suppose $\mathfrak{U}$ is a group variety. Let’s define $N_{\mathfrak{U}} \subset \mathbb{N}$ as a such set of numbers, that for any finite group $G$, $|G| \in N_{\mathfrak{U}}$ implies $G \in \...

**6**

votes

**1**answer

113 views

### Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups

Let $\left(W\text{, }S\right)$ be a Gromov hyperbolic Coxeter system and denote by $\partial W$ the corresponding Gromov boundary. For $z\in\partial W$ let $\alpha$, $\beta$ be infinite geodesic paths ...

**2**

votes

**1**answer

137 views

### Reduced expression and Bruhat order

For $n\ge 3$. Let $s_1\cdots s_n$ be a reduced expression of $x$. Suppose $s_1\cdots s_{n-1}\le w$ and $s_2\cdots s_{n}\le w$.
Does this imply $x\le w$?

**5**

votes

**1**answer

190 views

### Easy example of an infinite simple group with an embedding into a finitely presented group

I would like an easy example of an infinite simple group along with an embedding into a finitely presented group. I know that there are infinite simple finitely presented group such as Thompson’s ...

**4**

votes

**1**answer

139 views

### Where or how can I find matrix representatives of the conjugacy classes of Conway's group Co??

I would like to find ($24\times 24$) matrices representing the various conjugacy classes of Conway's group $\mathrm{Co}_0$ acting on the Leech lattice in the usual coordinate system given by the MOG. ...