Full paper in PDF:
$% A. J. Díaz-Barriga, F. González-Acuña, F. Marmolejo, and L. Román, Active sums I, Rev. Mat. Complut. 17 (2004), 287–319.%$

Active Sums I

Given a generating family of subgroups of a group , closed under conjugation and with partial order compatible with inclusion, a new group can be constructed, taking into account the multiplication in the subgroups and their mutual actions given by conjugation. The group is called the active sum of , has as a homomorph and is such that , where denotes the center.

The basic question we investigate in this paper is: when is the active sum of the family isomorphic to the group ?

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 with an odd prime, simple groups with trivial Schur multiplier, and special linear groups with a few exceptions.

We show as well that every finite group such that is not cyclic is the active sum of proper normal subgroups.

Key words: active sums, active sums of cyclic groups, regularity and independence, atomic and molecular groups.
2000 Mathematics Subject Classification: 20D99, 20E99, 20E15, 20J05, 08A55.