Lithuanian Mathematical Journal

, Volume 56, Issue 4, pp 492–502

The Influence of the Complete Nonexterior Square Graph on some Infinite Groups

Article

DOI: 10.1007/s10986-016-9331-2

Cite this article as:
Erwin, D. & Russo, F.G. Lith Math J (2016) 56: 492. doi:10.1007/s10986-016-9331-2
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Abstract

The notion of nonabelian exterior square may be formulated for a pro-p-group G (p prime), getting the complete nonabelian exterior square\( G\widehat{\varLambda}G \) of G. We introduce the complete nonexterior square graph\( {\widehat{\varGamma}}_G \) of G, investigating finiteness conditions on G from restrictions on \( {\widehat{\varGamma}}_G \) and viceversa. This graph has the set of vertices G − (G), where (G) is the set of all elements of G commuting with respect to the operator \( \widehat{\varLambda} \), and two vertices x and y are joined by an edge if \( x\widehat{\varLambda}y\ne 1 \). Studying \( {\widehat{\varGamma}}_G \), we find the well-known noncommuting graph as a subgraph. Moreover, we show results on the structure of G and introduce a new class of groups, which originates naturally when G is infinite but \( {\widehat{\varGamma}}_G \) is finite.

Keywords

complete nonabelian exterior square noncommuting graph pro-p-groups capable groups exterior degree 

MSC

05C25 20E18 20J05 05C63 

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa

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