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

Alejandro J. DíAZ-BARRIGA, Francisco GONZÁLEZ-ACUÑA,
Francisco MARMOLEJO, and Leopoldo ROMÁN
Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria
04510 México D.F. México

Received: June 26, 2003
Accepted: December 19, 2003
ABSTRACT

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  3
p  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/G 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.