Active sums. I.
Let \(U=\bigcup_iF_i\) be the disjoint union of a generating family \({\mathcal F}=\{F_i\}\) of subgroups of a group \(G\), closed under conjugation and with partial order compatible with inclusion. Define two binary operations on \(U\): partial multiplication on \(U\) induced by the multiplication in the subgroups and a global operation induced by conjugation in \(U\), \(x^y=y^{-1}xy\in F_k\) if \(x\in F_i\), \(y\in F_j\) and \(F^y_i=F_k\). Then \(U\) is a partial algebra called an active add determined by \(\mathcal F\). This active add determines a group \(S\) which is the image of \(U\) under left adjoint to the inclusion functor from the category of groups to the category of active adds. \(S\) is called the active sum of \(U\). The authors show that \(\varphi^{-1}(Z(G))=Z(S)\), where \(\varphi\) is the natural epimorphism from \(S\) onto \(G\) and \(Z\) denotes the center. They basically investigate when is the active sum \(S\) of the family \(\mathcal F\) isomorphic to the group \(G\) and show that the following groups are active sums of cyclic subgroups: free groups, groups of order cube of an odd prime, simple groups with trivial Schur multiplier, and special linear groups. They also show that every finite group \(G\) with \(G/G'\) non-cyclic is the active sum of proper normal subgroups.
scientific article; zbMATH DE number 2109938