Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2025-01-03T17:53:27.507Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [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]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 92 - 100
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was supported by NSF Grant DMS-1653002.

References

Besche, H. U., Eick, B. and O’Brien, E. A., ‘The groups of order at most 2000’, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1–4.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playout, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997), 235—265.CrossRefGoogle Scholar
Brauer, R. and Fowler, K. A., ‘On groups of even order’, Ann. of Math. 62 (1955), 567–583.CrossRefGoogle Scholar
Broshi, A. M., ‘Finite groups whose Sylow subgroups are abelian’, J. Algebra 17 (1971), 74–82.CrossRefGoogle Scholar
Costanzo, D. G. and Lewis, M. L., ‘The cyclic graph of a $2$ -Frobenius group’, Preprint, 2021, arXiv:2103.15574.CrossRefGoogle Scholar
Giudici, M. and Parker, C., ‘There is no upper bound for the diameter of the commuting graph of a finite group’, J. Combin. Theory Ser. A 120 (2013), 1600–1603.CrossRefGoogle Scholar
Giudici, M. and Pope, A., ‘On bounding the diameter of the commuting graph of a group’, J. Group Theory 17 (2014), 131–149.CrossRefGoogle Scholar
Iranmanesh, A. and Jafarzadeh, A., ‘On the commuting graph associated with the symmetric and alternating groups’, J. Algebra Appl. 7 (2008), 129–146.CrossRefGoogle Scholar
Morgan, G. L. and Parker, C. W., ‘The diameter of the commuting graph of a finite group with trivial centre, J. Algebra 393 (2013), 41–59.CrossRefGoogle Scholar
Parker, C., ‘The commuting graph of a soluble group’, Bull. Lond. Math. Soc. 45 (2013), 839–848.CrossRefGoogle Scholar
Segev, Y. and Seitz, G. M., ‘Anisotropic groups of type ${A}_n$ and the commuting graph of finite simple groups’, Pacific J. Math. 202 (2002), 125–225.CrossRefGoogle Scholar
Solomon, R. and Woldar, A., ‘Simple groups are characterized by their non-commuting graphs’, J. Group Theory 16 (2013), 793–824.CrossRefGoogle Scholar
Taunt, D. R., ‘On $A$ -groups’, Math. Proc. Cambridge Philos. Soc. 45 (1949), 24–42.CrossRefGoogle Scholar
The GAP Group, GAP—Groups, algorithms, and programming, version 4.11.0, 2020, http://www.gap-system.org/.Google Scholar