Classification and properties of the π\pi-submaximal subgroups in minimal nonsolvable groups

Let $\pi$ be a set of primes. According to H. Wielandt, a subgroup $H$ of a finite group $X$ is called a $\pi$-submaximal subgroup if there is a monomorphism $\phi:X\rightarrow Y$ into a finite group $Y$ such that $X^\phi$ is subnormal in $Y$ and $H^\phi=K\cap X^\phi$ for a $\pi$-maximal subgroup $K$ of $Y$. In his talk at the well-known conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the $\pi$-submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set $\pi$ of primes, we obtain a description of the $\pi$-submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt's problem

Existence criterion for Hall subgroups of finite groups

In the paper we obtain an existence criterion for Hall subgroups of finite groups in terms of a composition series.Comment: We made some editor corrections in the tex

Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture

Let $G$ be a locally finite group and $F(G)$ the Hirsch--Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G/F(G)$ in $G$. Assume that there is a number $k$ such that the length of every chain of nested centralizers in $G$ does not exceed $k$. The Borovik--Khukhro conjecture states, in particular, that under this assumption the quotient $G/S$ contains an abelian subgroup of index bounded in terms of $k$. We disprove this statement and prove some its weaker analog

On generations by conjugate elements in almost simple groups with socle \mbox{}^2F_4(q^2)'

We prove that if L=\mbox{}^2F_4(2^{2n+1})' and $x$ is a nonidentity automorphism of $L$ then $G=\langle L,x\rangle$ has four elements conjugate to $x$ that generate $G$. This result is used to study the following conjecture about the $\pi$-radical of a finite group: Let $\pi$ be a proper subset of the set of all primes and let $r$ be the least prime not belonging to $\pi$. Set $m=r$ if $r=2$ or $3$ and set $m=r-1$ if $r\geqslant 5$. Supposedly, an element $x$ of a finite group $G$ is contained in the $\pi$-radical $\operatorname{O}_\pi(G)$ if and only if every $m$ conjugates of $x$ generate a $\pi$-subgroup. Based on the results of this paper and a few previous ones, the conjecture is confirmed for all finite groups whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, or unitary simple group, or to one of the groups of type ${}^2B_2(2^{2n+1})$, ${}^2G_2(3^{2n+1})$, ${}^2F_4(2^{2n+1})'$, $G_2(q)$, or ${}^3D_4(q)$

On embedding theorems for X\mathfrak{X}-subgroups

Let $\mathfrak{X}$ be a class of finite groups closed under subgroups, homomorphic images, and extensions. We study the question which goes back to the lectures of H. Wielandt in 1963-64: For a given $\mathfrak{X}$-subgroup $K$ and maximal $\mathfrak{X}$-subgroup $H$, is it possible to see embeddability of $K$ in $H$ (up to conjugacy) by their projections onto the factors of a fixed subnormal series. On the one hand, we construct examples where $K$ has the same projections as some subgroup of $H$ but is not conjugate to any subgroup of $H$. On the other hand, we prove that if $K$ normalizes the projections of a subgroup $H$, then $K$ is conjugate to a subgroup of $H$ even in the more general case when $H$ is a submaximal $\mathfrak{X}$-subgroup