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/G' is not cyclic is the active sum of proper normal subgroups.
@article{urn:eudml:doc:44380,
title = {Active sums I.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {17},
year = {2004},
pages = {287-319},
zbl = {1069.20046},
mrnumber = {MR2083957},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44380}
}
Díaz-Barriga, J. Alejandro; González-Acuña, Francisco; Marmolejo, Francisco; Román, Leopoldo. Active sums I.. Revista Matemática de la Universidad Complutense de Madrid, Tome 17 (2004) pp. 287-319. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44380/