To any finite group G, we may associate a graph whose vertices are the elements of G and where two distinct vertices x and y are adjacent if and only if the order of the subgroup generated by them is divisible by at least 3 distinct primes. We prove that the subgraph of this graph induced by the non-isolated vertices is connected and has diameter at most 5.
THE NON-TWO-PRIMES GRAPH OF A FINITE GROUP
Lucchini A.
2026
Abstract
To any finite group G, we may associate a graph whose vertices are the elements of G and where two distinct vertices x and y are adjacent if and only if the order of the subgroup generated by them is divisible by at least 3 distinct primes. We prove that the subgraph of this graph induced by the non-isolated vertices is connected and has diameter at most 5.
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