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Bulletin of the Australian Mathematical Society Bulletin of the Australian Mathematical Society

Article contents

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 [Opens in a new window] ,
MARK L. LEWIS [Opens in a new window] ,
STEFANO SCHMIDT [Opens in a new window] ,
EYOB TSEGAYE [Opens in a new window]  and
GABE UDELL [Opens in a new window]
DAVID G. COSTANZO
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA e-mail: dcostan2@kent.edu
MARK L. LEWIS
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
STEFANO SCHMIDT
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA e-mail: sas2393@columbia.edu
EYOB TSEGAYE
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA e-mail: eytsegay@stanford.edu
GABE UDELL
Affiliation:
Department of Mathematics, Pomona College, Claremont, CA 91711, USA e-mail: grua2017@mymail.pomona.edu
Corresponding
E-mail address:
dcostan2@kent.edu
sas2393@columbia.edu
eytsegay@stanford.edu
grua2017@mymail.pomona.edu
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} \lbrace 1\rbrace $ 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 graph enhanced power graph Z-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 , First View , pp. 1 - 7
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

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