On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group

Abstract

Let $\pi$ be a proper subset of the set of prime numbers. Denote by $r$ the least prime not contained in $\pi$ and set $m=r$ for $r=2$ and $3$ and $m=r-1$ for $r\geqslant5$. The conjecture under consideration claims that a conjugacy class $D$ of a finite group $G$ generates a $\pi$-subgroup of $G$ (equivalently, is contained in the $\pi$-radical) if and only if any $m$ elements of $D$ generate a $\pi$-group. It is shown that this conjecture holds if every non-Abelian composition factor of $G$ is isomorphic to a sporadic, an alternating, a linear, or a unitary simple group. Bibliography: 49 titles.

On generations by conjugate elements in almost simple groups with socle 2𝐹4(𝑞2)′

“…The main result of [10] constitutes finding explicit, albeit not always best possible, upper bounds on α S (x) for all nonabelian simple groups S. These bounds and their refinements have been extensively used in applications of the classification of finite simple groups. For example, they are substantially used in proofs of various analogues of the famous Baer-Suzuki theorem, see [3,5,6,7,8,9,17,18,19,16,15]. For practical use, the estimates on α S (x) from [10] are not always sufficient.…”