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We study the commuting graph of n n matrices over the field of p-adics Qₚ, whose vertices are non-scalar n n matrices with entries in Qₚ and whose edges connect pairs of matrices that commute under matrix multiplication. We prove that this graph is connected if and only if n 3, with n neither prime nor a power of p. We also prove that in the case of p=2 and n=2q for q a prime with q 7, the commuting graph has the maximum possible diameter of 6; these are the first known such examples independent of the axiom of choice. We also find choices of p and n yielding diameter 4 and diameter 5 commuting graphs, and prove general bounds depending on p and n.
Ralph Morrison (Thu,) studied this question.