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Maximal linear groups induced on the Frattini quotient of a $p$-group | | John Bamberg
; S. P. Glasby
; Luke Morgan
; Alice C. Niemeyer
; | | Date: |
17 Mar 2016 | | Abstract: | Let $p>3$ be a prime. For each maximal subgroup $H leqslant
mathrm{GL}(d,p)$ with $|H|=p^{mathrm O(d^2)}$, we construct a $d$-generator
finite $p$-group $G$ with the property that $mathrm{Aut}(G)$ induces $H$ on
the Frattini quotient $G/Phi(G)$ and $|G|= p^{mathrm O(d^4)}$. A significant
feature of this construction is that $|G|$ is very small compared to $|H|$,
shedding new light upon a celebrated result of Bryant and Kov’acs. The groups
$G$ that we exhibit have exponent $p$, and of all such groups $G$ with the
desired action of $H$ on $G/Phi(G)$, the construction yields groups with
smallest nilpotency class, and in most cases, the smallest order. | | Source: | arXiv, 1603.5384 | | Services: | Forum | Review | PDF | Favorites | |
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