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

Article contents

THE NUMBER OF SET ORBITS OF PERMUTATION GROUPS AND THE GROUP ORDER

Part of: Permutation groups

Published online by Cambridge University Press:  27 January 2022

MICHAEL GINTZ Open the ORCID record for MICHAEL GINTZ[Opens in a new window] ,
THOMAS M. KELLER Open the ORCID record for THOMAS M. KELLER[Opens in a new window] ,
MATTHEW KORTJE Open the ORCID record for MATTHEW KORTJE[Opens in a new window] ,
ZILI WANG Open the ORCID record for ZILI WANG[Opens in a new window]  and
YONG YANG Open the ORCID record for YONG YANG[Opens in a new window]
MICHAEL GINTZ
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ08544, USA e-mail: mgintz@princeton.edu
THOMAS M. KELLER
Affiliation:
Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX78666, USA e-mail: keller@txstate.edu
MATTHEW KORTJE
Affiliation:
Department of Science and Mathematics, Cedarville University, 251 N Main St, Cedarville, OH45314, USA e-mail: mkortje@cedarville.edu
ZILI WANG
Affiliation:
Department of Mathematics, University of California-Berkeley, 970 Evans Hall, Berkeley, CA94720-3840, USA e-mail: ziliwang271@berkeley.edu
YONG YANG*
Affiliation:
Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX78666, USA
*
e-mail: yang@txstate.edu
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Abstract

If G is permutation group acting on a finite set $\Omega $ , then this action induces a natural action of G on the power set $\mathscr{P}(\Omega )$ . The number $s(G)$ of orbits in this action is an important parameter that has been used in bounding numbers of conjugacy classes in finite groups. In this context, $\inf ({\log _2 s(G)}/{\log _2 |G|})$ plays a role, but the precise value of this constant was unknown. We determine it where G runs over all permutation groups not containing any ${{\textrm {A}}}_l, l> 4$ , as a composition factor.

Keywords

permutation groupsset orbitsgroup order

MSC classification

Primary: 20B05: General theory for finite groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 1 , August 2022 , pp. 89 - 101
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research was supported by NSF-REU grant DMS-1757233 and NSA grant H98230-21-1-0333. Y. Yang was also partially supported by a grant from the Simons Foundation (#499532).

References

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