# Properties of subgroups not containing their centralizers

Lemnouar Noui^{[1]}

- [1] Department of Mathematics Faculty of Science University of Batna, Algeria

Annales mathématiques Blaise Pascal (2009)

- Volume: 16, Issue: 2, page 267-275
- ISSN: 1259-1734

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topNoui, Lemnouar. "Properties of subgroups not containing their centralizers." Annales mathématiques Blaise Pascal 16.2 (2009): 267-275. <http://eudml.org/doc/10579>.

@article{Noui2009,

abstract = {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 $N_\{c\}(G)$ the number of conjugacy classes and $N_\{a\}$ the number of abelian maximal subgroups and $N_\{na\}$ the number of nonabelian maximal subgroups.},

affiliation = {Department of Mathematics Faculty of Science University of Batna, Algeria},

author = {Noui, Lemnouar},

journal = {Annales mathématiques Blaise Pascal},

keywords = {Maximal subgroup; divisible groups; p-groups; center; conjugacy classes; finite -groups; semi-direct products; numbers of conjugacy classes; numbers of maximal subgroups},

language = {eng},

month = {7},

number = {2},

pages = {267-275},

publisher = {Annales mathématiques Blaise Pascal},

title = {Properties of subgroups not containing their centralizers},

url = {http://eudml.org/doc/10579},

volume = {16},

year = {2009},

}

TY - JOUR

AU - Noui, Lemnouar

TI - Properties of subgroups not containing their centralizers

JO - Annales mathématiques Blaise Pascal

DA - 2009/7//

PB - Annales mathématiques Blaise Pascal

VL - 16

IS - 2

SP - 267

EP - 275

AB - 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 $N_{c}(G)$ the number of conjugacy classes and $N_{a}$ the number of abelian maximal subgroups and $N_{na}$ the number of nonabelian maximal subgroups.

LA - eng

KW - Maximal subgroup; divisible groups; p-groups; center; conjugacy classes; finite -groups; semi-direct products; numbers of conjugacy classes; numbers of maximal subgroups

UR - http://eudml.org/doc/10579

ER -

## References

top- Michael Reid, The number of conjugacy classes, Amer. Math. Monthly 105 (1998), 359-361 Zbl0924.20015MR1614889
- Derek J. S. Robinson, Finiteness conditions and generalized soluble groups. Part 1, (1972), Springer-Verlag, New York Zbl0243.20032MR332989
- Derek J. S. Robinson, A course in the theory of groups, 80 (1996), Springer-Verlag, New York Zbl0836.20001MR1357169
- Gary Sherman, A lower bound for the number of conjugacy classes in a finite nilpotent group, Pacific J. Math. 80 (1979), 253-254 Zbl0377.20017MR534714
- N. Watson, Subgroups of finite abelian groups, (1995)

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