On the sharp Baer--Suzuki theorem for the $pi$-radical

Let $pi$ be a set of primes such that $|pi|geqslant 2$ and $pi$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $pi$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $rgeqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in $Opi(G)$ if and only if every $m$ elements of $D$ generate a $pi$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to alternating, linear and unitary simple groups.

On an open problem of Skiba

Let $sigma={sigma_{i}|iin I}$ be some partition of the set $mathbb{P}$ of all primes, that is, $mathbb{P}=bigcup_{iin I}sigma_{i}$ and $sigma_{i}cap sigma_{j}=emptyset$ for all $i eq j$. Let $G$ be a finite group. A set $mathcal {H}$ of subgroups of $G$ is said to be a complete Hall $sigma$-set of $G$ if every non-identity member of $mathcal {H}$ is a Hall $sigma_{i}$-subgroup of $G$ and $mathcal {H}$ contains exactly one Hall $sigma_{i}$-subgroup of $G$ for every $sigma_{i}in sigma(G)$. $G$ is said to be a $sigma$-group if it possesses a complete Hall $sigma$-set. A $sigma$-group $G$ is said to be $sigma$-dispersive provided $G$ has a normal series $1 = G_1<G_2<cdots< G_t< G_{t+1} = G$ and a complete Hall $sigma$-set ${H_{1}, H_{2}, cdots, H_{t}}$ such that $G_iH_i = G_{i+1}$ for all $i= 1,2,ldots t$. In this paper, we give a characterizations of $sigma$-dispersive group, which give a positive answer to an open problem of Skiba in the paper.