# Active sums I.

J. Alejandro Díaz-Barriga; Francisco González-Acuña; Francisco Marmolejo; Leopoldo Román

Revista Matemática Complutense (2004)

- Volume: 17, Issue: 2, page 287-319
- ISSN: 1139-1138

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topDíaz-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 -

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