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4 votes
1 answer
266 views

Are (group theoretic) Markov properties on groups with decidable word problems, decidable?

(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 ...
6 votes
3 answers
872 views

An element $g$ in a group such that neither $g=1$ nor $g\ne 1$ can be proved.

Edited (this question contains two versions of a similar question) Is there some finitely presented group $G$ generated by $g_1,...,g_n$ such that there is an element $g\in G$ expressed as a finite ...
4 votes
2 answers
544 views

Membership problem in monoids

What is the simplest example of a monoid with undecidable membership problem? In other words, I'm looking for a concrete monoid $S$ such that there is no algorithm which takes elements $s_1,...,s_n$ ...
8 votes
1 answer
338 views

How bad can the recursive properties of finitely presented groups be?

Any finitely presented group naturally gives rise to an edge-labeled graph (the Cayley graph) and I am considering paths through this graph. Paths correspond to infinite sequences of generators, so ...
0 votes
0 answers
106 views

A decision problem of an inverse problem in finite group theory

A finite group $G$ is called integral if there is a finite group $H$ such that $G\cong H'$. In Araujo, Cameron, Casolo, Matucci's paper, integrals of groups, they tried to solve a problem as following:...
26 votes
2 answers
1k views

Is the cohomology ring of a finite group computable?

Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\mathbb{...
17 votes
0 answers
969 views

Groups generated by 3 involutions

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...