Some properties of groups which allow homomorphisms onto their direct square (Q1330030)

The author calls a nontrivial group \(G\) an M.B. group if \(G \times G\) is a homomorphic image of \(G\). The first construction of a finitely presented M.B. group was obtained by Baumslag and Miller in 1988. A general method generalizing the previous construction was given by Hirshon and Meier in 1992. The aim of this paper is to gather some properties of M.B. groups. For example: Corollary 2. A necessary and sufficient condition for a non-trivial group \(G\) to be an M.B. group is that there exists a homomorphic image \(A\) of \(G\) such that \(G\) is also a homomorphic image of \(A\) and \(A^ 2 \cong A\). Corollary 6. There exists a finitely generated group \(A\) such that \(A\) contains a finitely presented M.B. group \(G\) as a direct factor with \(G\) containing as a subgroup an isomorphic copy of any finitely presented group and such that \(A^ 2 \cong A\).