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# Questions tagged [gr.group-theory]

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

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### 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 ...
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### 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 ...
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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 $G \in \... 1answer 113 views ### Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups Let$\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$be infinite geodesic paths ... 1answer 137 views ### Reduced expression and Bruhat order For$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$? 1answer 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 ... 1answer 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. ...

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