Properties of subgroups not containing their centralizers
Annales mathématiques Blaise Pascal, Volume 16 (2009) no. 2, pp. 267-275.

In this paper, we give a generalization of Baer Theorem on the injective property of divisible abelian groups. As consequences of the obtained result we find a sufficient condition for a group $G$ to express as semi-direct product of a divisible subgroup $D$ and some subgroup $H$. We also apply the main Theorem to the $p$-groups with center of index ${p}^{2}$, for some prime $p$. For these groups we compute ${N}_{c}\left(G\right)$ the number of conjugacy classes and ${N}_{a}$ the number of abelian maximal subgroups and ${N}_{na}$ the number of nonabelian maximal subgroups.

DOI: 10.5802/ambp.266
Classification: 14L05, 20D25, 20K27, 20E28
Keywords: Maximal subgroup, divisible groups, p-groups, center, conjugacy classes
Lemnouar Noui 1

1 Department of Mathematics Faculty of Science University of Batna, Algeria
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Lemnouar Noui. Properties of subgroups not containing their centralizers. Annales mathématiques Blaise Pascal, Volume 16 (2009) no. 2, pp. 267-275. doi : 10.5802/ambp.266. https://ambp.centre-mersenne.org/articles/10.5802/ambp.266/

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[3] Derek J. S. Robinson A course in the theory of groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York, 1996 | MR | Zbl

[4] Gary Sherman A lower bound for the number of conjugacy classes in a finite nilpotent group, Pacific J. Math., Volume 80 (1979) no. 1, pp. 253-254 | MR | Zbl

[5] N. Watson Subgroups of finite abelian groups, 1995 (Summer research paper, Haverford College)

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