05/26/2021, 10:31 AM
PROGRESS UPDATE (May 26, 2021):
Found some more bookmarks at MO Is there an infinite group with exactly two conjugacy classes?.
The first is an "higly non trivial" result by Osin (2010)
Theorem 1.1 (pag. 2) Any countable group G can be embedded into a 2-generated group C such that any two elements of the same order are conjugate in C
Strengthening Higman-Neumann-Neumann previous embedding theorem
that is, modulo some details, given a countable group G of function we can add new function (extend the group) to obtain a new group where we can always solve the superfunction equation. Not only that you could think that all those groups are pretty boring and there are not many of them. Instead we have
In the MO answer by Dan Sălăjan there is an interesting note that could be interesting if we look at superfunction-closed groups as groups where, in some sense, there exist infinite chains of hyperoperations.
Correction to the previous post: if G is finite and is super-closed then it has to be the unique group with two element.
https://proofwiki.org/wiki/Finite_Group_...2_Elements
Found some more bookmarks at MO Is there an infinite group with exactly two conjugacy classes?.
The first is an "higly non trivial" result by Osin (2010)
Theorem 1.1 (pag. 2) Any countable group G can be embedded into a 2-generated group C such that any two elements of the same order are conjugate in C
Strengthening Higman-Neumann-Neumann previous embedding theorem
Quote:HNN Embedding thm any countable group G can be embedded into a countable group B such that every two elements of the same order are conjugate in B
that is, modulo some details, given a countable group G of function we can add new function (extend the group) to obtain a new group where we can always solve the superfunction equation. Not only that you could think that all those groups are pretty boring and there are not many of them. Instead we have
Quote:Corollary 1.3 (pag. 2) There exists an uncountable set of pairwise nonisomorphictorsion-free2-generated groups with exactly2conjugacy classes.
In the MO answer by Dan Sălăjan there is an interesting note that could be interesting if we look at superfunction-closed groups as groups where, in some sense, there exist infinite chains of hyperoperations.
Quote:As a psychological curiosity, Per Enflo writes in his Autobiography that the existence of groups with two conjugacy classes was a key insight behind his many solutions to outstanding problems in Functional Analysis.*Boldface is mine.
"I made important progress in mathematics in 1966, but it was more on the level of new insights, than actual results. When thinking about topological groups in the spirit of Hilbert's fifth problem (I had gradually modified the Hibert problem to some very general program: To decide whether different classes of topological groups shared properties with Lie groups) I was wondering whether there exist "very non-commutative" groups i.e. groups, where all elements except e, are conjugate to each other. I constructed such groups, by finding the right finite phenomenon and then make an induction. I understood, that this is a very general construction scheme (or "philosophy"), that probably could be applied to various infinite or infinite-dimensional problems*. And actually – this philosophy is behind several of my best papers - the solution of the basis and approximation problem, the solution of the invariant subspace problem for Banach spaces, the solution of Smirnov's problem on uniform embeddings into Hilbert space and more." (Dan Sălăjan, MO, 2013)
Correction to the previous post: if G is finite and is super-closed then it has to be the unique group with two element.
https://proofwiki.org/wiki/Finite_Group_...2_Elements
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)