New posts in Combinatorial Group Theory

When two elements of a group generate the same subgroup

Can the map sending a presentation to its group be considered as a functor?

Which one of the following groups is isomorphic to the group $G\ $?

Let $G$ be a group with a free subgroup of rank $2$. Let $H\leq G$ be such that $[G:H]<\infty$. Then $H$ also contains a free subgroup of rank $2$.

Presentation for the product of two cyclic groups

One missing step in proving $\mathbb{Z}\times \mathbb{Z} \cong \langle a,b\,|\, [a,b]=1\rangle$

Find the group with presentation $\langle x, y: xy=yx, x^5=y^3\rangle$.

Finitely generated group unlikely to be generated by randomly chosen elements.

How to approach proofs similar to "Show a group, $G$, is infinite if $G = \langle r, s, t\mid rst = 1\rangle $"

Word metric on a finitely generated subgroup versus the word metric of its finitely generated parent