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Finite groups in which subnormalizers are subgroups

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Finite groups in which subnormalizers are subgroups RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA CARLOCASOLO Finite groups in which subnormalizers are subgroups Rendiconti del Seminario Matematico della Università di Padova, tome 82 (1989), p. 25-53. <http://www.numdam.org/item?id=RSMUP_1989__82__25_0> © Rendiconti del Seminario Matematico della Università di Padova, 1989, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est consti- tutive 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/ Finite Groups in which Subnormalizers are Subgroups. CARLO CASOLO (*) Let H be a subgroup of a group G. We put and call it the « subnormalizer » of H in G (see [7 ; p. 238]). In gen- eral, Ba(H) is not a subgroup (see [7]). The aim of this, paper is to study the class of groups (which we call sn-groups) in which the sub- normalizer of every subgroup is a subgroup. As observed in [7; p. 238], is a subgroup of G if and only if H is subnormal in U, V~, whenever and Furthermore, if G is finite and Ba(H) G then, by a subnormality criterion of H. Wielandt [10], .g is subnormal in Ba(H); thus SG(H) is the maximal subgroup of G in which g is embedded as a subnormal subgroup. From now on, « group » will mean « finite group ». In the first section of this paper we show that the property of being an sn-group has a local character. Namely, we define for every prime p, the class of sn (p)-groups of those groups in which the sub- normalizer of every p-subgroup is a subgroup, and prove that G is an sn-group if and only if G is an s

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