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ON $p$-GROUPS WITH AUTOMORPHISM GROUPS RELATED TO THE CHEVALLEY GROUP $G_{2}(p)$

Part of: Representation theory of groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  08 January 2020

JOHN BAMBERG Open the ORCID record for JOHN BAMBERG [Opens in a new window] ,
SAUL D. FREEDMAN Open the ORCID record for SAUL D. FREEDMAN [Opens in a new window]  and
LUKE MORGAN
JOHN BAMBERG
Affiliation:
Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia email john.bamberg@uwa.edu.au
SAUL D. FREEDMAN*
Affiliation:
Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia email sdf8@st-andrews.ac.uk
LUKE MORGAN
Affiliation:
Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia email luke.morgan@famnit.upr.si
*
✉ email sdf8@st-andrews.ac.uk
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Abstract

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\text{Aut}(P)$ induces $N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$ on the Frattini quotient $P/\unicode[STIX]{x1D6F7}(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p=3$ our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_{2}(q)$ on the octonion algebra over $\mathbb{F}_{q}$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_{q}[G_{2}(q)]$-module.

Keywords

p-groupexterior squareG2(q)

MSC classification

Primary: 20C33: Representations of finite groups of Lie type
Secondary: 20G15: Linear algebraic groups over arbitrary fields 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 3 , June 2020 , pp. 321 - 331
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author acknowledges the support of the Australian Research Council Future Fellowship FT120100036. The second author was supported by a Hackett Foundation Alumni Honours Scholarship, a Hackett Postgraduate Research Scholarship, and an Australian Government Research Training Program Scholarship at The University of Western Australia. The third author was supported by the Australian Research Council grant DE160100081.

†

Current address: School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK.

‡

Current address: UP FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia. Also affiliated with: UP IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia.

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