The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the r …

What is the length of the shortest word $w\in F_2$ such that $w(x,y)$ is trivial for every $x,y\in S_n$? There is a simple argument showing that we must have $\ell(w)\geq n$. See …

Suppose I have a finitely presented group $G$. By this, I mean I know explicitly what $S$ and $R$ are such that $G = \langle S \mid R \rangle$. Suppose I have a subgroup generated …

Let $BS(m,n)$ be the Baumslag-Solitar group defined by $B(m,n) = < a,b ~|~ b a^m b^{-1} = a^n > $, $mn \neq 0$. There is a linear representation of $BS(m,n)$ by mapping $a$ to …

The Novikov--Boone Theorem, which is perhaps the archetypal local unsolvability result in group theory, states existence of a finitely presented group whose word problem is recursi …

A group $G$ is co-hopfian if every injection $f\colon G \rightarrow G$ is an automorphism, or equivalently if $G$ is not isomorphic to any of its proper subgroups. Miller and Schu …

Suppose we have a finitely presented group $G$ with a concrete presentation and a subgroup $H$, generated by a finite set of elements from $G$. How to find the presentation for $H$ …

Consider a finitely presented group $G$ with presentation $P$ given by $\left\langle g_1,\ldots,g_n|\, r_1,\ldots,r_m\right\rangle$, equipped with a homomorphism $\rho\colon\, G\to …

Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is full if it acts as a nonidenti …

Hello! I have a problem which i thought was really easy to solve but now Iam here =) I need to construct a final combination of a content based on combinations of various sub-con …

I learned from MO http://mathoverflow.net/questions/46115/subgroups-of-a-finite-abelian-group that the problem of enumerating subgroups (not up to isomorphism) of finite abelian gr …

Let $G$ be the finitely presented group with two generators $a,b$ and one relation $b^2=1$. First question: Does that group have a name ? Perhaps an answer to this question c …

This maybe a very general question. If we have a group given by its presentation only, what kind of properties could be proven about it? I know examples about non-amenability of …

Let $G$ be a free group. Then $G/G^{(n)}$ ($G^{(n)}$ is the $n$th derived subgroup.) acts on $G^{(n)}/G^{(n+1)}$ by conjugation, which makes $G^{(n)}/G^{(n+1)}$ a $\mathbb{Z}[G/G^{ …

I have met the following problem. A group $G$ is given as follows $G = \langle x,y,t| y^{-2}xy^2 = x,t^{-1}yt =y^2 ,t^{-1}xt = xy^{-1}xy\rangle$ Is the subgroup generated by $y$ …