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Register a new userClassification and properties of the $pi$-submaximal subgroups in minimal nonsolvable groups
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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 $phi:Xrightarrow Y$ into a finite group $Y$ such that $X^phi$ is subnormal in $Y$ and $H^phi=Kcap X^phi$ 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 Wielandts problem.
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We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group, in general very largely in terms of the dimension.
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