Bulletin of the
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ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2016; 53(5): 1353-1362

Online first article August 25, 2016      Printed September 30, 2016

https://doi.org/10.4134/BKMS.b150636

Copyright © The Korean Mathematical Society.

Commuting elements with respect to the operator $\wedge $ in infinite groups

Rashid Rezaei and Francesco G. Russo

Malayer University, Private Bag X1, 7701 Rondebosch

Abstract

Using the notion of complete nonabelian exterior square $G \widehat{\wedge} G$ of a pro-$p$-group $G$ ($p$ prime), we develop the theory of the exterior degree $\widehat{\mathrm{d}}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs $\mathrm{d}(G)$. Among the main results of this paper, we describe upper and lower bounds for $\widehat{\mathrm{d}}(G)$ with respect to $\mathrm{d}(G)$. Here the size of the second homology group $H_2(G,\mathbb{Z}_p)$ (over the $p$-adic integers $\mathbb{Z}_p$) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of $H_2(G,\mathbb{Z}_p)$ both on $G$ and $\widehat{\mathrm{d}}(G)$.

Keywords: exterior degree, exterior center, exterior centralizers, complete nonabelian exterior square

MSC numbers: Primary 20J05, 20J06; Secondary 20E10, 20P05

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