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THE CYCLIC GRAPH OF A Z-GROUP

Part of: Graph theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  14 December 2020

DAVID G. COSTANZO Open the ORCID record for DAVID G. COSTANZO [Opens in a new window] ,
MARK L. LEWIS Open the ORCID record for MARK L. LEWIS [Opens in a new window] ,
STEFANO SCHMIDT Open the ORCID record for STEFANO SCHMIDT [Opens in a new window] ,
EYOB TSEGAYE Open the ORCID record for EYOB TSEGAYE [Opens in a new window]  and
GABE UDELL Open the ORCID record for GABE UDELL [Opens in a new window]
DAVID G. COSTANZO
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH44242, USA e-mail: dcostan2@kent.edu
MARK L. LEWIS*
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH44242, USA
STEFANO SCHMIDT
Affiliation:
Department of Mathematics, Columbia University, New York, NY10027, USA e-mail: sas2393@columbia.edu
EYOB TSEGAYE
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA94305, USA e-mail: eytsegay@stanford.edu
GABE UDELL
Affiliation:
Department of Mathematics, Pomona College, Claremont, CA91711, USA e-mail: grua2017@mymail.pomona.edu
*
e-mail: lewis@math.kent.edu
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Abstract

For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G\setminus {\{\,1\,\}} $ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle $ is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta (G)$ for a Z-group G.

Keywords

cyclic graphenhanced power graphZ-group

MSC classification

Primary: 20F16: Solvable groups, supersolvable groups
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 2 , October 2021 , pp. 295 - 301
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

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