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On \(p\)-groups with automorphism groups related to the Chevalley group \(G_2(p)\). (English) Zbl 1484.20016

Summary: Let \(p\) be an odd prime. We construct a \(p\)-group \(P\) of nilpotency class two, rank seven and exponent \(p\), such that \(\operatorname{Aut}(P)\) induces \(N_{\mathrm{GL}(7,p)}(G_2(p)) = Z (\mathrm{GL}(7,p)) G_2(p)\) on the Frattini quotient \(P/ \Phi (P)\). 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.

MSC:

20C33 Representations of finite groups of Lie type
20G40 Linear algebraic groups over finite fields
20D15 Finite nilpotent groups, \(p\)-groups
20D45 Automorphisms of abstract finite groups
20F28 Automorphism groups of groups

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References:

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