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 − Ẑ(G), where Ẑ(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.
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The author was supported in part by the grant CSUR 93652 of NRF (South Africa). (Francesco G. Russo)
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Erwin, D., Russo, F.G. The Influence of the Complete Nonexterior Square Graph on some Infinite Groups. Lith Math J 56, 492–502 (2016). https://doi.org/10.1007/s10986-016-9331-2
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DOI: https://doi.org/10.1007/s10986-016-9331-2