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Register a new userA polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group
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We show that, there exists a constant $a$ such that, for every subgroup $H$ of a finite group $G$, the number of maximal subgroups of $G$ containing $H$ is bounded above by $a|G:H|^{3/2}$. In particular, a transitive permutation group of degree $n$ has at most $an^{3/2}$ maximal systems of imprimitivity. When $G$ is soluble, generalizing a classic result of Tim Wall, we prove a much stroger bound, that is, the number of maximal subgroups of $G$ containing $H$ is at most $|G:H|-1$.
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The purpose of this paper is to prove that if $G$ is a transitive permutation group of degree $ngeq 2$, then $G$ can be generated by $lfloor cn/sqrt{log{n}}rfloor$ elements, where $c:=sqrt{3}/2$. Owing to the transitive group $D_8circ D_8$ of degree $8$, this upper bound is best possible. Our new result improves a 2018 paper by the author, and makes use of the recent classification of transitive groups of degree $48$.
A graph $Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $Aut(Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $Aut(Ga)$ then $Ga$ is called a normal Cayley graph of $T$. Let $r$ be an odd prime. Fang et al. cite{FMW} proved that, with a finite number of exceptions for finite simple group $T$, every connected symmetric Cayley graph of $T$ of valency $r$ is normal. In this paper, employing maximal factorizations of finite almost simple groups, we work out a possible list of those exceptions for $T$.
Building on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver-Watkins classes of edge-transitive maps.
The $2$-closure $overline{G}$ of a permutation group $G$ on $Omega$ is defined to be the largest permutation group on $Omega$, having the same orbits on $OmegatimesOmega$ as $G$. It is proved that if $G$ is supersolvable, then $overline{G}$ can be found in polynomial time in $|Omega|$. As a byproduct of our technique, it is shown that the composition factors of $overline{G}$ are cyclic or alternating of prime degree.
Denote by $m(G)$ the largest size of a minimal generating set of a finite group $G$. We estimate $m(G)$ in terms of $sum_{pin pi(G)}d_p(G),$ where we are denoting by $d_p(G)$ the minimal number of generators of a Sylow $p$-subgroup of $G$ and by $pi(G)$ the set of prime numbers dividing the order of $G$.
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