Characterizing finite <i>p</i>-groups by their Schur multipliers

Peyman Niroomand

doi:10.1016/j.crma.2012.10.018

Group Theory

Characterizing finite p-groups by their Schur multipliers
[Caractérisation des p-groupes finis par leurs multiplicateurs de Schur]

Présenté par : the Editorial Board

Peyman Niroomand ¹

¹ School of Mathematics and Computer Science, Damghan University, Damghan, Iran

Comptes Rendus. Mathématique, Volume 350 (2012) no. 19-20, pp. 867-870.

Résumé
Abstract

Il est montré dans J.A. Green (1956) [5] que pour tout p-groupe dʼordre $p^{n}$ on a $| M (G) | = p^{\frac{n (n - 1)}{2} - t (G)}$ où $t (G) ⩾ 0$ . Dans Ya.G. Berkovich (1991) [1], G. Ellis (1999) [4], et X. Zhou (1994) [14] la structure de G a été classifiée par plusieurs auteurs pour $t (G) = 0, 1, 2, 3$ . Également, dans A.R. Salemkar et al. (2007) [12] la structure de G est caractérisée lorsque $t (G) = 4$ et $Z (G)$ est abelien élémentaire, mais il y a quelques trous dans la classification complète de ces groupes. Cette Note est consacrée à la caractérisation de la structure de G lorsque $t (G) = 4$ , sans restriction aucune et dʼune manière différente, plus directe que les approches de Ya.G. Berkovich (1991) [1], G. Ellis (1999) [4], A.R. Salemkar et al. (2007) [12], et X. Zhou (1994) [14].

It has been proved in J.A. Green (1956) [5] for every p-group of order $p^{n}$ , $| M (G) | = p^{\frac{1}{2} n (n - 1) - t (G)}$ , where $t (G) ⩾ 0$ . In Ya.G. Berkovich (1991) [1], G. Ellis (1999) [4], and X. Zhou (1994) [14], the structure of G has been characterized for $t (G) = 0, 1, 2, 3$ by several authors. Also in A.R. Salemkar et al. (2007) [12], the structure of G characterized when $t (G) = 4$ and $Z (G)$ is elementary abelian, but there are some missing points in classifying the structure of these groups. This paper is devoted to classify the structure of G when $t (G) = 4$ without any condition and with a short and quite different way to that of Ya.G. Berkovich (1991) [1], G. Ellis (1999) [4], A.R. Salemkar et al. (2007) [12], and X. Zhou (1994) [14].

Reçu le : 2012-05-11
Accepté le : 2012-10-15
Publié le : 2012-10-01

DOI : 10.1016/j.crma.2012.10.018

Affiliations des auteurs :

Peyman Niroomand ¹

¹ School of Mathematics and Computer Science, Damghan University, Damghan, Iran

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     title = {Characterizing finite \protect\emph{p}-groups by their {Schur} multipliers},
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     pages = {867--870},
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Peyman Niroomand. Characterizing finite p-groups by their Schur multipliers. Comptes Rendus. Mathématique, Volume 350 (2012) no. 19-20, pp. 867-870. doi : 10.1016/j.crma.2012.10.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.10.018/

Bibliographie
Cité par

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[11] P. Niroomand; R. Rezaei On the exterior degree of finite groups, Comm. Algebra, Volume 39 (2011), pp. 1-9

[12] A.R. Salemkar; M.R.R. Moghaddam; M. Davarpanah; F. Saeedi A remark on the Schur multiplier of p-groups, Comm. Algebra, Volume 35 (2007), pp. 1215-1221

[13] E. Schenkman Group Theory, Robert E. Krieger Publishing Co., Huntington, NY, 1975 (corrected reprint of the 1965 edition)

[14] X. Zhou On the order of Schur multipliers of finite p-groups, Comm. Algebra, Volume 1 (1994), pp. 1-8

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