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EXTENDING RESULTS OF MORGAN AND PARKER ABOUT COMMUTING GRAPHS

Part of: Structure and classification of infinite or finite groups Graph theory

Published online by Cambridge University Press:  11 May 2021

NICOLAS F. BEIKE Open the ORCID record for NICOLAS F. BEIKE [Opens in a new window] ,
RACHEL CARLETON Open the ORCID record for RACHEL CARLETON [Opens in a new window] ,
DAVID G. COSTANZO Open the ORCID record for DAVID G. COSTANZO [Opens in a new window] ,
COLIN HEATH Open the ORCID record for COLIN HEATH [Opens in a new window] ,
MARK L. LEWIS Open the ORCID record for MARK L. LEWIS [Opens in a new window] ,
KAIWEN LU Open the ORCID record for KAIWEN LU [Opens in a new window]  and
JAMIE D. PEARCE Open the ORCID record for JAMIE D. PEARCE [Opens in a new window]
NICOLAS F. BEIKE
Affiliation:
Department of Mathematics and Statistics, 501 Lincoln Building, Youngstown State University, Youngstown, OH 44555, USA e-mail: nfbeike@student.ysu.edu
RACHEL CARLETON
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA e-mail: rcarlet3@kent.edu
DAVID G. COSTANZO
Affiliation:
School of Mathematical and Statistical Sciences, O-110 Martin Hall, Box 340975, Clemson University, Clemson, SC 29634, USA e-mail: dcostan2@kent.edu
COLIN HEATH
Affiliation:
Department of Mathematics, 3620 S. Vermont Ave., KAP 104, University of Southern California, Los Angeles, CA 90089, USA e-mail: colinjhe@usc.edu
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: kailu@umich.edu
JAMIE D. PEARCE
Affiliation:
Department of Mathematics, University of Texas at Austin, 2515 Speedway, PMA 8.100, Austin, TX 78712, USA e-mail: jamie.pearce@utexas.edu
*
e-mail: lewis@math.kent.edu
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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.

Keywords

A-groupcommuting graphFrobenius group

MSC classification

Primary: 20E34: General structure theorems
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 92 - 100
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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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