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Random bases for coprime linear groups

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 Added by Zolt\\'an Halasi
 Publication date 2019
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and research's language is English




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The minimal base size $b(G)$ for a permutation group $G$, is a widely studied topic in the permutation group theory. Z. Halasi and K. Podoski proved that $b(G)leq 2$ for coprime linear groups. Motivated by this result and the probabilistic method used by T. C. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber that for coprime linear groups $Gleq GL(V)$, whether there exists a constant $c$ such that the probability of that a random $c$-tuple is a base for $G$ tends to 1 as $|V|toinfty$. While the answer to this question is negative in general, it is positive under the additional assumption that $G$ is even primitive as a linear group. In this paper, we show that almost all $11$-tuples are bases for coprime primitive linear groups.



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Let $G$ be a finite group admitting a coprime automorphism $alpha$ of order $e$. Denote by $I_G(alpha)$ the set of commutators $g^{-1}g^alpha$, where $gin G$, and by $[G,alpha]$ the subgroup generated by $I_G(alpha)$. We study the impact of $I_G(alpha)$ on the structure of $[G,alpha]$. Suppose that each subgroup generated by a subset of $I_G(alpha)$ can be generated by at most $r$ elements. We show that the rank of $[G,alpha]$ is $(e,r)$-bounded. Along the way, we establish several results of independent interest. In particular, we prove that if every element of $I_G(alpha)$ has odd order, then $[G,alpha]$ has odd order too. Further, if every pair of elements from $I_G(alpha)$ generates a soluble, or nilpotent, subgroup, then $[G,alpha]$ is soluble, or respectively nilpotent.
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Let G be a linear group acting on the finite vector space V and assume that (|G|,|V|)=1. In this paper we prove that G has a base size at most two and this estimate is sharp. This generalizes and strengthens several former results concerning base sizes of coprime linear groups. As a direct consequence, we answer a question of I. M. Isaacs in the affirmative. Via large orbits this is related to the k(GV) theorem.
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