[Les groupes finis métacycliques comme sommes actives de sous-groupes cycliques]
La notion de somme active fournit un analogue pour les groupes de ce qu'est la somme directe pour les groupes abéliens. Une question naturelle est alors de déterminer quels groupes sont la somme active d'une famille de sous-groupes cycliques. De nombreux groupes possèdent cette propriété, mais la question demeurait ouverte pour les groupes finis métacycliques. Dans cette note, nous montrons que tout groupe fini métacyclique s'obtient comme la somme active d'une famille discrète de sous-groupes cycliques.
The notion of active sum provides an analogue for groups of what the direct sum is for abelian groups. One natural question then is which groups are the active sum of a family of cyclic subgroups. Many groups have been found to give a positive answer to this question, while the case of finite metacyclic groups remained unknown. In this note we show that every finite metacyclic group can be recovered as the active sum of a discrete family of cyclic subgroups.
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@article{CRMATH_2014__352_7-8_567_0,
author = {D{\'\i}az-Barriga, Alejandro and Gonz\'alez-Acu\~na, Francisco and Marmolejo, Francisco and Romero, Nadia},
title = {Finite metacyclic groups as active sums of cyclic subgroups},
journal = {Comptes Rendus. Math\'ematique},
pages = {567--571},
publisher = {Elsevier},
volume = {352},
number = {7-8},
year = {2014},
doi = {10.1016/j.crma.2014.06.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2014.06.006/}
}
TY - JOUR AU - Díaz-Barriga, Alejandro AU - González-Acuña, Francisco AU - Marmolejo, Francisco AU - Romero, Nadia TI - Finite metacyclic groups as active sums of cyclic subgroups JO - Comptes Rendus. Mathématique PY - 2014 SP - 567 EP - 571 VL - 352 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.06.006/ DO - 10.1016/j.crma.2014.06.006 LA - en ID - CRMATH_2014__352_7-8_567_0 ER -
%0 Journal Article %A Díaz-Barriga, Alejandro %A González-Acuña, Francisco %A Marmolejo, Francisco %A Romero, Nadia %T Finite metacyclic groups as active sums of cyclic subgroups %J Comptes Rendus. Mathématique %D 2014 %P 567-571 %V 352 %N 7-8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.06.006/ %R 10.1016/j.crma.2014.06.006 %G en %F CRMATH_2014__352_7-8_567_0
Díaz-Barriga, Alejandro; González-Acuña, Francisco; Marmolejo, Francisco; Romero, Nadia. Finite metacyclic groups as active sums of cyclic subgroups. Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 567-571. doi : 10.1016/j.crma.2014.06.006. http://www.numdam.org/articles/10.1016/j.crma.2014.06.006/
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