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THE CYCLIC GRAPH OF A Z-GROUP

Published online by Cambridge University Press:  14 December 2020

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

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.

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

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