2021/03/12 by Nicolas F. Beike, Beike, Nicolas F., Rachel Carleton +11
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary 20D40 Secondary 05C25 #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2103.07355
openalex publication_date 2021/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Morgan and Parker have proved that if G is a group satisfying the condition that Z(G) = 1, then the connected components of the commuting graph of G have diameter at most 10. Parker has proved that if in addition G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a 2-Frobenius group, and if the commuting graph of G is connected, then its diameter is at most 8. We prove that the hypothesis Z (G) = 1 in these results can be replaced with G' ∩ Z(G) = 1. We also prove that if G is solvable and G/Z(G) is either a Frobenius group or a 2-Frobenius group, then the commuting graph of G is disconnected.