Active Sums I

Given a generating family F of subgroups of a group G, closed under conjugation and with partial order compatible with inclusion, a new group S can be constructed, taking into account the multiplication in the subgroups and their mutual actions given by conjugation. The group S is called the active sum of F, has G as a homomorph and is such that S/Z(S) ' G/Z(G), where Z denotes the center. The basic question we investigate in this paper is: when is the active sum S of the family F isomorphic to the group G? The conditions found to answer this question are often of a homological nature. We show that the following groups are active sums of cyclic subgroups: free groups, semidirect products of cyclic groups, Coxeter groups, Wirtinger approximations, groups of order p3 with p an odd prime, simple groups with trivial Schur multiplier, and special linear groups SLn(q) with a few exceptions. We show as well that every finite group G such that G/G0 is not cyclic is the active sum of proper normal subgroups