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

  • JOHN BAMBERG (a1), SAUL D. FREEDMAN (a2) and LUKE MORGAN (a3)

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.

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Copyright

Corresponding author

Footnotes

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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.

Footnotes

References

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

  • JOHN BAMBERG (a1), SAUL D. FREEDMAN (a2) and LUKE MORGAN (a3)

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