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Register a new userAn upper bound on the Chebotarev invariant of a finite group
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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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Let $cO_K$ be a complete discrete valuation ring of residue characteristic $p>0$, and $G$ be a finite flat group scheme over $cO_K$ of order a power of $p$. We prove in this paper that the Abbes-Saito filtration of $G$ is bounded by a simple linear function of the degree of $G$. Assume $cO_K$ has generic characteristic 0 and the residue field of $cO_K$ is perfect. Fargues constructed the higher level canonical subgroups for a Barsotti-Tate group $cG$ over $cO_K$ which is not too supersingular. As an application of our bound, we prove that the canonical subgroup of $cG$ of level $ngeq 2$ constructed by Fargues appears in the Abbes-Saito filtration of the $p^n$-torsion subgroup of $cG$.
We study the class of finite groups $G$ satisfying $Phi (G/N)= Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe to all finite groups and answer in the affirmative a long-standing question of Christensen whether the class of finite groups which possess complements for each of their normal subgroups is subnormally closed.
Let $G$ be an amenable group. We define and study an algebra $mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of $G$. For a just infinite amenable group $G$, we show that $mathcal{A}_{sn}(G)$ is nilpotent if and only if $G$ is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $operatorname{rad} ell^1(G)^{**}$ for an amenable branch group $G$, and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely-generated counterexamples to a question of Dales and Lau, first resolved by the author in a previous article, which asks whether we always have $(operatorname{rad} ell^1(G)^{**})^{Box 2} = { 0 }$. We further study this question by showing that $(operatorname{rad} ell^1(G)^{**})^{Box 2} = { 0 }$ imposes certain structural constraints on the group $G$.
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 $|G:H|-1$.
A group $G$ is said to be $frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 in G$, there exists $y in G$ such that $G = langle x_1, y rangle = langle x_2, y rangle$. In other words, $s(G) geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.
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