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A tensor product approach to compute $2$ -nilpotent multiplier of $p$ -groups

F. Fasihi

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

E-mail Address: faez.fasihi@yahoo.com

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and

S. Hadi Jafari

https://orcid.org/0000-0002-1166-0173

Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

E-mail Address: s.hadi_jafari@yahoo.com

Corresponding author.

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https://doi.org/10.1142/S1793557122500905Cited by:4 (Source: Crossref)

Abstract

Let $G$ be a group given by a free presentation $G ≅ F / R$ . The 2-nilpotent multiplier of $G$ is the abelian group $𝕄^{(2)} (G) = (R \cap γ_{3} (F)) / γ_{3} (R, F)$ which is invariant of $G$ [R. Baer, Representations of groups as quotient groups, I, II, and III, Trans. Amer. Math. Soc.58 (1945) 295–419]. An effective approach to compute the 2-nilpotent multiplier of groups has been proposed by Burns and Ellis [On the nilpotent multipliers of a group, Math. Z.226 (1997) 405–428], which is based on the nonabelian tensor product. We use this method to determine the explicit structure of $𝕄^{(2)} (G)$ , when $G$ is a finite (generalized) extra special $p$ -group. Moreover, the descriptions of the triple tensor product $\otimes^{3} G$ , and the triple exterior product $\land^{3} G$ are given.

Communicated by A. F. Vasilev

Keywords:

AMSC: Primary: 20D15, Primary: 20C25, Secondary: 20E34

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