Let $G$ be a finite group and $\phi : G \rightarrow G$ be a group automorphism such that for more than $\frac{3}{4}$ of elements $g \in G$ we have$$\phi(g) = g^{-1}$$Prove that for all $g \in G$$$\...

This question concerns the mixed Braid groups $B_{n,P}$. Suppose $P$ is a partition of $\{1, \ldots, n\}$. Consider the subgroup $S_{n,P}$ of the symmetric group $S_n$ consisting of the permutations ...

During my research in Algebraic Geometry, I was led to the following problem in Combinatorial Group Theory, strictly related to finite quotients of pure surface braid groups.Let $G$ be a finite group....

Let $F$ be the free profinite group of rank $r$. Let $\tilde{F}$ be the free profinite group of rank $r + 1$, and $C$ be a subgroup of $\tilde{F}$ which is, of course non-canonically, isomorphic to $\...

Given a right-angled Artin group (RAAG) $A_\Gamma$ it is well-known that two generators $u$ and $v$ commute if and only if $u$ and $v$ are connected with an edge in $\Gamma$. But, is there are general ...

Let $n$ be a positive integer, and consider the hypersurface of singular $n\times n$ matrices over $\mathbb{F}_2$, denoted$$\mathcal{S}_n = \{X\in M_n(\mathbb{F}_2) : \det(X)=0\}.$$Note that\...

Consider the following one-relator group: $$G = \langle x,y,w,z \mid x^3[x,y][w,z] = 1\rangle$$ where $[a,b] = aba^{-1}b^{-1}$ denotes the commutator. It is the free product with amalgamation $$F(x,y) ...

Let $w$ be a word uniformly sampled from the cyclically reduced word in the free group on $r$ elements.I'm looking for the expected length of the Whitehead minimization of $w$.I don't need a precise ...

I have the following variant of the birthday problem / coupon collector problem:Let $w$ be a reduced word sampled uniformly from the set of reduced words of length $n$ in the free group $F_r$. As $n$...

Are there known examples of naturally-occurring groups where the word problem is algorithmically solvable but not easily? I ask because I'm looking at word problems of some groups of interest to me, ...

Let $I$ be a finite category and $F \colon I \rightarrow \mathrm{Grp}$ be a functor into the category of groups, such that $F(c)$ is a finite group for every $c \in Obj(I)$.Question. Does $\mathrm{...

Let $F_n$ be the free group on $n$ generators where $n \geq 2$. Let's say I have a sequence of words $W=\{w_i\}_{i=0}^\infty$ such that:$w_i \sqsubseteq w_j$ ($w_i$ is a subword of $w_j$) iff $i=j$....

In 1965, Murasugi [1] conjectured that any finitely presented group with deficiency at least two has trivial centre. The year before, he had proved it true for one-relator groups, and in [1] he proved ...

Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \...

(Link to SE duplicate: https://math.stackexchange.com/questions/4959071/are-group-theoretic-markov-properties-on-groups-with-decidable-word-problems)The Adian-Rabin theorem says that if a property of ...