Anton Klyachko
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 Nov 9 comment Explicitly showing that a free group is LERF Surely, there is an algorithm for finding this finite-index subgroup and the free complement in this subgroup. Just look at the proof of Hall's free-factor theorem. It is quite easy using Stallings graphs.. Nov 9 comment Explicitly showing that a free group is LERF Pablo, every f.g. subgroup $M$ of a f.g. free group $F$ is a free factor of a finite-index subgroup of $F$. So, the problem is reduced to the case where $M$ is a free factor of $F$, where it is quite easy. Nov 9 comment Fantastic properties of Z/2Z I added these features, thanks, @Sam. Nov 9 revised Fantastic properties of Z/2Z added 189 characters in body Nov 9 comment Does the linear automorphism group determine the vector space? Thanks, @Todd. I corrected the number. Nov 9 revised Does the linear automorphism group determine the vector space? edited body Nov 9 comment Fantastic properties of Z/2Z @Sam, I bet your single identity forms a basis of identities of the group. (This means that all other identities are consequences of this one.) Nov 9 revised Does the linear automorphism group determine the vector space? added 192 characters in body Nov 8 revised Does the linear automorphism group determine the vector space? added 23 characters in body Nov 8 answered Does the linear automorphism group determine the vector space? Nov 3 comment Fantastic properties of Z/2Z @Denis, of course not. The infinite dihedral group $Z/2Z*Z/2Z$ has elements of infinite order. Nov 3 comment Fantastic properties of Z/2Z @Todd, I understand you but I prefer to use slightly different terminology. See this discussion in comments: mathoverflow.net/q/92972/24165 Nov 3 comment Fantastic properties of Z/2Z @Emil, every group of exponent two is a direct product of several two-element groups (if you believe the Axiom of Choice). Nov 3 awarded Nice Answer Nov 3 comment Which groups are LERF? It is not LERF because of the Mikhailova subgroup with undecidable membership problem. Oct 31 awarded Necromancer Oct 9 comment Two questions about axiomatic rank of groups @M.Shahryari, no way. Consider the variety of abelian groups. It satisfies the metabelian law $[[x,y],[z,t]]=1$ or the nilpotency law $[[x,y],z]=1$. Do you believe that these laws are equivalent to two-variable laws? Oct 8 comment Two questions about axiomatic rank of groups To question 1 the answer is obviously negative. Just consider the trivial variety, it has finite axiomatic rank but satisfies all identities. Aug 27 comment Are infinite groups “locally topologizable”? It is on page 339 of English edition. Unfortunately, Google preview rejects to show this page to me (but I may read the table of contents). Aug 26 comment Are infinite groups “locally topologizable”? I have a Russian edition. This is Chapter 10, Section 31, Subsection 3.