Active sums I.

Díaz-Barriga, J. Alejandro, et al. "Active sums I.." Revista Matemática Complutense 17.2 (2004): 287-319. <http://eudml.org/doc/44380>.

@article{Díaz2004,
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 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.},
author = {Díaz-Barriga, J. Alejandro, González-Acuña, Francisco, Marmolejo, Francisco, Román, Leopoldo},
journal = {Revista Matemática Complutense},
keywords = {Grupos finitos; Grupos cíclicos; Subgrupos; active sums of cyclic groups; partial algebras; regularity; independence; atomic groups; molecular groups; categories of groups},
language = {eng},
number = {2},
pages = {287-319},
title = {Active sums I.},
url = {http://eudml.org/doc/44380},
volume = {17},
year = {2004},
}

TY - JOUR
AU - Díaz-Barriga, J. Alejandro
AU - González-Acuña, Francisco
AU - Marmolejo, Francisco
AU - Román, Leopoldo
TI - Active sums I.
JO - Revista Matemática Complutense
PY - 2004
VL - 17
IS - 2
SP - 287
EP - 319
AB - 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.
LA - eng
KW - Grupos finitos; Grupos cíclicos; Subgrupos; active sums of cyclic groups; partial algebras; regularity; independence; atomic groups; molecular groups; categories of groups
UR - http://eudml.org/doc/44380
ER -