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As the "second-generation" proof of the Classification of Finite Simple Groups is being written up in the volumes by Gorenstein, Lyons, Aschbacher, Smith, Solomon, and others (see e.g. this question) …

As indicated in the comments, it's undecidable in general to take as input a finite presentation of a group and try to output whether or not this group is free or not. This is a direct consequence of …

It is well-known that one of the corollaries of the classification of finite simple groups (CFSG) is that every finite simple group can be generated by two elements. In a comment on an answer to an ol …

Yes. This is given by OEIS sequence A160870. The number of subgroups of index $n$ in $\mathbf{Z}^m$ is there denoted $T(n,m)$. There is a recursive formula in terms of the divisors of $n$ given at thi …

In combinatorial group theory, loosely speaking almost any problem one can imagine, in full generality, turns out to be undecidable. This includes the word problem, the isomorphism problem, the trivia …

Let $G = \langle A \mid R \rangle$ be a finitely presented group, given by a finite presentation. If $G$ is abelian, then we can verify this fact: simply verify the fact that $[a, b] = 1$ for all gene …

Obviously, there are lots of answers already, but I thought I'd give the proof of (5) as given by Jordan already in 1870 in his Traité des substitutions -- this has the benefit of being quite clear to …

For a finitely presented group $G$, generated by a finite set $A$, the commutator problem is the decision problem: given a word $w$ over the alphabet $A \cup A^{-1}$, can one decide if $w$ is a commut …

As mentioned in the comments, this is still considered an open problem. I thought I'd flesh out a few aspects. A solution was claimed in 1992 by Juhasz, but it seems to have failed to convince experts …

As you mention in your update, you have a general answer, but if you want a concrete answer for the low-dimensional integral cohomology of $G = \operatorname{GL}(3,2)$ (or any other finite group!), yo …

Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this dir …

Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area. First, in "Yang-M …

Novikov's centrally-symmetric group $\mathfrak{A}_P$ is a torsion-free group with undecidable word problem, constructed in [1]. Novikov did not prove it is torsion-free but, as Adian points out in [Ad …

Let $d$ be a square-free positive integer, and let $\mathcal{O}_d$ be the ring of integers of the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$. Consider the Bianchi group $\Gamma_d = \oper …

Is the conjugacy problem for two-relator groups known to be undecidable? The word problem for two-relator groups is a famous open problem (appearing e.g. as Question 9.29 in the Kourovka notebook), an …