Finite groups in \(\beta\mathbb{N}\) are trivial (Q1362895)

A group \(G\) endowed with a topology \(\tau\) is called a left topological group if all left translations of \(G\) are continuous. A topological space \(X\) with a distinguished element \(e\) and a partial binary operation is called a local left topological group if there exists a left topological group \(G\) such that \(e\) is the identity of \(G\), \(X\) is an open neighbourhood of \(e\) in \(G\) and the operation of \(G\) extends the operation on \(X\). Then \(\beta X\) is the Stone-Čech compactification of \(X\) with the discrete topology and \(\beta X\) is considered with a partial binary operation which is a natural extension of the operation on \(X\). Let \(\tilde {X}\) denote the set of all free ultrafilters on \(X\) converging to \(e\), then \(\tilde {X}\) is a semigroup. The following results are announced: If \(G\) is a countable torsion free abelian group and if \(K\) is a finite subgroup of \(\beta G\), then there exist an elementary local left topological group \(X\) and a homomorphism \(f:X\to K\) such that the extension \(\tilde {f}:\tilde {X}\to K\) of \( f\) is a retraction. If \(X\) and \(Y\) are elementary local left topological groups such that there exist homomorphisms \(f:X\to K\) and \(g:X\to K\) for a finite group \(K\), then there exists an isomorphism \(h:X\to Y\) with \(f=g\circ h\). The semigroups of ultrafilters of countable non-discrete metrizable groups are topologically isomorphic. Any finite subgroup in \(\beta G\) of a countable torsion free abelian group \(G\) is trivial.