Motived by an intrinisic necessary condition for the purifiability of subgroups of primary abelian groups due to K. Benabdallah and T. Okuyama we introduce new functors on the category of pairs (G, A),where A is a subgroup of G,to the category of Z/pZ-vector spaces. The vanishing of these functors leads to the notion of vertical subgroup which is a weakening of purity but also an essential component of the latter. In fact, a vertical subgroup is pure if and only if it is neat. We establish various facts about vertical subgroups and “maximal” vertical subgroups and apply the resulting theory to the problem of purifiability. We show that the class of quasi-complete groups is precisely the class of reduced groups in which every subgroup satisfying the intrinsic necessary condition for purifiability is in fact purifiable. This is also the class of reduced p-groups in which the maximal vertical subgroups are precisely the pure subgroups.

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Étude sur les sous-groupes d'un groupe abélien Bulletin de la S. M. F., tome 88 (1960), p. 217-227 © Bulletin de la S. M. F., 1960, tous droits réservés. L'accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ BULL.

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