Minimal generation of transitive permutation groups

It is proved in [21] that there is a constant $c$ such that each transitive permutation group of degree $d\ge 2$ can be generated by $\lfloor cd/\sqrt{\log{d}}\rfloor$ elements. In this paper, we explicitly estimate $c$

Finite groups with large Chebotarev invariant

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ is said to invariably generate $G$ 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 authors recently showed that for each $\epsilon>0$, there exists a constant $c_{\epsilon}$ such that $C(G)\le (1+\epsilon)\sqrt{|G|}+c_{\epsilon}$. This bound is asymptotically best possible. In this paper we prove a partial converse: namely, for each $\alpha>0$ there exists an absolute constant $\delta_{\alpha}$ such that if $G$ is a finite group and $C(G)>\alpha\sqrt{|G|}$, then $G$ has a section $X/Y$ such that $|X/Y|\geq \delta_{\alpha}\sqrt{|G|}$, and $X/Y\cong \mathbb{F}_q\rtimes H$ for some prime power $q$, with $H\le \mathbb{F}_q^{\times}$

Invariable generation of permutation and linear groups

A subset {x 1,x 2,…,x d} of a group G invariably generates G if {x 1 g 1 ,x 2 g 2 ,…,x d g d } generates G for every d-tuple (g 1,g 2…,g d)∈G d. We prove that a finite completely reducible linear group of dimension n can be invariably generated by ⌊[Formula presented]⌋ elements. We also prove tighter bounds when the field in question has order 2 or 3. Finally, we prove that a transitive [respectively primitive] permutation group of degree n≥2 [resp. n≥3] can be invariably generated by O([Formula presented]) [resp. O([Formula presented])] elements. </p