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EXCEPTIONAL GROUPS OF ORDER $p^6$ FOR PRIMES $p\geq 5$

Part of: Permutation groups Representation theory of groups

Published online by Cambridge University Press:  24 February 2025

E. A. O’BRIEN Open the ORCID record for E. A. O’BRIEN [Opens in a new window] ,
SUNIL KUMAR PRAJAPATI Open the ORCID record for SUNIL KUMAR PRAJAPATI [Opens in a new window]  and
AYUSH UDEEP Open the ORCID record for AYUSH UDEEP [Opens in a new window]
E. A. O’BRIEN
Affiliation:
Department of Mathematics, University of Auckland, Auckland, New Zealand e-mail: [email protected]
SUNIL KUMAR PRAJAPATI
Affiliation:
Indian Institute of Technology Bhubaneswar, Arugul Campus, Jatni, Khurda 752050, India e-mail: [email protected]
AYUSH UDEEP*
Affiliation:
Indian Institute of Science Education and Research Mohali, Sector 81, SAS Nagar, Punjab 140306, India
*
e-mail: [email protected]
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Abstract

The minimal faithful permutation degree $\mu (G)$ of a finite group G is the least integer n such that G is isomorphic to a subgroup of the symmetric group $S_n$. If G has a normal subgroup N such that $\mu (G/N)> \mu (G)$, then G is exceptional. We prove that the proportion of exceptional groups of order $p^6$ for primes $p \geq 5$ is asymptotically zero. We identify $(11p+107)/2$ such groups and conjecture that there are no others.

Keywords

exceptional groupsgroups of order p6minimal degree permutation representations

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups
Secondary: 20B05: General theory for finite groups 20C15: Ordinary representations and characters
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

O’Brien was supported by the Marsden Fund of New Zealand Grant 23-UOA-080 and by a Research Award of the Alexander von Humboldt Foundation. Prajapati acknowledges the Science and Engineering Research Board, Government of India, for financial support through grant MTR/2019/000118. Udeep thanks the Indian Institute of Science Education and Research Mohali for his Postdoctoral Fellowship.

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