Classification and properties of the $$\pi $$ -submaximal subgroups in minimal nonsolvable groups
Tóm tắt
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 celebrated 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.
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