# COMMUTING ELEMENTS WITH RESPECT TO THE OPERATOR Λ IN INFINITE GROUPS

• Rezaei, Rashid (Department of Mathematics Malayer University) ;
• Russo, Francesco G. (Department of Mathematics and Applied Mathematics University of Cape Town)
• Published : 2016.09.30

#### Abstract

Using the notion of complete nonabelian exterior square $G\hat{\wedge}G$ of a pro-p-group G (p prime), we develop the theory of the exterior degree $\hat{d}(G)$ in the infinite case, focusing on its relations with the probability of commuting pairs d(G). Among the main results of this paper, we describe upper and lower bounds for $\hat{d}(G)$ with respect to 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 $\hat{d}(G)$.

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#### Cited by

1. Strong subgroup commutativity degree and some recent problems on the commuting probabilities of elements and subgroups vol.39, pp.8, 2016, https://doi.org/10.2989/16073606.2016.1247118
2. The Influence of the Complete Nonexterior Square Graph on some Infinite Groups vol.56, pp.4, 2016, https://doi.org/10.1007/s10986-016-9331-2