On the class of groups with pronormal hall π-subgroups

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

“…Evidently, m π (G) ⊆ sm π (G) for any group G. The inverse inclusion does not hold in general as one can see in the above example: any Sylow 2-subgroup of P SL 2 (7) is {2, 3}-submaximal but is not {2, 3}-maximal.…”

“…In general, the intersection of a π-maximal subgroup H with a normal subgroup N of G is not a π-maximal subgroup of N. For example, it is easy to show that a Sylow 2-subgroup H of G = P GL 2 (7) is {2, 3}-maximal in G but H ∩ N / ∈ m {2,3} (N) for N = P SL 2 (7).…”

“…This statement has many consequences for arbitrary finite groups: criteria of the existence [29] and the conjugacy [28], closeness of the class of D π -groups under taking extensions, some analog of the Frattini argument for Hall subgroups [23] etc. The strongest results on Hall subgroups turned out to be formulated in term of the pronormality [7,[31][32][33]: the pronormality of Hall subgroups in the finite simple groups, the existence of a pronormal π-subgroup in any group containing π-Hall subgroups etc.…”

“…We denote by Hall π (G) the set of π-Hall subgroups of a group G. The D π -groups and the groups containing π-Hall subgroups are well investigated (see a survey [30] and next works [7,9,23,33]). In particular, it is proved that G ∈ D π if and only if every composition factor of G is a D π -groups (see [21,Theorem 7.7], [30,Theorem 6.6], [6, Chapter 2, Theorem 6.15]) and the simple D π -groups are described in [19, Theorem 3] (see also [30,Theorem 6.9]).…”

The existence of pronormal π-Hall subgroups in E π-groups

“…In [4] some properties were specified of the groups such that π-Hall subgroups exist and all are pronormal.…”