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EXTENDING RESULTS OF MORGAN AND PARKER ABOUT COMMUTING GRAPHS

Published online by Cambridge University Press:  11 May 2021

NICOLAS F. BEIKE
Affiliation:
Department of Mathematics and Statistics, 501 Lincoln Building, Youngstown State University, Youngstown, OH 44555, USA e-mail: [email protected]
RACHEL CARLETON
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA e-mail: [email protected]
DAVID G. COSTANZO
Affiliation:
School of Mathematical and Statistical Sciences, O-110 Martin Hall, Box 340975, Clemson University, Clemson, SC 29634, USA e-mail: [email protected]
COLIN HEATH
Affiliation:
Department of Mathematics, 3620 S. Vermont Ave., KAP 104, University of Southern California, Los Angeles, CA 90089, USA e-mail: [email protected]
MARK L. LEWIS*
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
KAIWEN LU
Affiliation:
Department of Mathematics, 2074 East Hall, 530 Church Street, University of Michigan, Ann Arbor, MI 48109, USA e-mail: [email protected]
JAMIE D. PEARCE
Affiliation:
Department of Mathematics, University of Texas at Austin, 2515 Speedway, PMA 8.100, Austin, TX 78712, USA e-mail: [email protected]

Abstract

Morgan and Parker proved that if G is a group with ${\textbf{Z}(G)} = 1$ , then the connected components of the commuting graph of G have diameter at most $10$ . Parker 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 $G' \cap {\textbf{Z}(G)} = 1$ . We also prove that if G is solvable and $G/{\textbf{Z}(G)}$ is either a Frobenius group or a $2$ -Frobenius group, then the commuting graph of G is disconnected.

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported by NSF Grant DMS-1653002.

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