The diameters of commuting graphs of linear groups and matrix rings over the integers modulo m
Abstract
The commuting graph of a group G, denoted by Gamma(G), is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. Let Z_m be the commutative ring of equivalence classes of integers modulo m. In this paper we investigate the connectivity and diameters of the commuting graphs of GL(n,Z_m) to contribute to the conjecture that there is a universal upper bound on diam(Gamma(G)) for any finite group G when Gamma(G) is connected. For any composite m, it is shown that Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) are connected and diam(Gamma(GL(n,Z_m))) = diam(Gamma(M(n,Z_m))) = 3. For m a prime, the instances of connectedness and absolute bounds on the diameters of Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) when they are connected are concluded from previous results.
Cite
@article{arxiv.1003.0156,
title = {The diameters of commuting graphs of linear groups and matrix rings over the integers modulo m},
author = {Michael Giudici and Aedan Pope},
journal= {arXiv preprint arXiv:1003.0156},
year = {2010}
}
Comments
10 pages, v.3: introduction & bibliography corrected