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Register a new userSharp upper bounds on the minimal number of elements required to generate a transitive permutation group
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Gareth Tracey
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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 $48$.
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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 $|G:H|-1$.
In this paper we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. We give bounds on the order, base size, minimal degree, fixity, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we classify finite semiprimitive groups that induce the alternating or symmetric group on the set of orbits of an intransitive normal subgroup.
A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved that a cycle base of a permutation group of degree $n$ can be constructed in polynomial time in~$n$.
We give upper bounds on the order of the automorphism group of a simple graph
A subset ${g_1, ldots , g_d}$ of a finite group $G$ invariably generates $G$ if the set ${g_1^{x_1}, ldots, g_d^{x_d}}$ generates $G$ for every choice of $x_i in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. The first author recently showed that $C(G)le betasqrt{|G|}$ for some absolute constant $beta$. In this paper we show that, when $G$ is soluble, then $beta$ is at most $5/3$. We also show that this is best possible. Furthermore, we show that, in general, for each $epsilon>0$ there exists a constant $c_{epsilon}$ such that $C(G)le (1+epsilon)sqrt{|G|}+c_{epsilon}$.
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