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

Abstract

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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 n a the number of nonabelian maximal subgroups.

How to cite

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Noui, 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

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  1. Michael Reid, The number of conjugacy classes, Amer. Math. Monthly 105 (1998), 359-361 Zbl0924.20015MR1614889
  2. Derek J. S. Robinson, Finiteness conditions and generalized soluble groups. Part 1, (1972), Springer-Verlag, New York Zbl0243.20032MR332989
  3. Derek J. S. Robinson, A course in the theory of groups, 80 (1996), Springer-Verlag, New York Zbl0836.20001MR1357169
  4. 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
  5. N. Watson, Subgroups of finite abelian groups, (1995) 

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