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Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of H G , where H G is the normal closure of H in G, that is, the smallest normal subgroup of G containing H. A group G is called an HNE2-group if all cyclic subgroups of order 2 and 4 of G are Hall normally embedded in G. In this paper, we prove that all HNE2-groups are 2-nilpotent. Furthermore, we also characterize the structure of finite group all of whose maximal subgroups are HNE2-groups. Finally, we determine finite non-solvable groups all of whose second maximal subgroups are HNE2-groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and σ = { σ i | i ∈ I } be a partition of the set of all primes P , that is, P = ∪ i ∈ I σ i and σ i ∩ σ j = ∅ for all i ≠ j . The natural numbers n and m are called σ-coprime if σ (n) ∩ σ (m) = ∅ . The group G is said to be: σ-primary if G is a σi-group for some i ∈ I ; σ-soluble if either G = 1 or every chief factor of G is σ-primary. A subgroup H of G is called σ-subnormal in G if there is a subgroup chain H = H 0 ≤ H 1 ≤ ⋯ ≤ H t = G such that either H i − 1 is normal in Hi or H i / (H i − 1) H i is σ-primary for all i = 1 , ... , t . In this paper, we show that G is σ-soluble provided G satisfies the following conditions: (1) G = A 1 A 2 = A 1 A 3 = A 2 A 3 , where A1, A2, A3 are all σ-soluble; (2) the three indices | G : N G (A 1 N σ ) | , | G : N G (A 2 N σ ) | , | G : N G (A 3 N σ ) | are pairwise σ-coprime. And we prove that if G is σ-soluble and A is a subgroup of G, then A is σ-subnormal in G if and only if | A B | σ i divides | G | σ i for every Hall σi-subgroup B of G and all σ i ∈ σ (G) . We also state and prove a σ-nilpotency criterion for G and a characterization of the σ-Fitting subgroup of G, which are related to this observation. [ABSTRACT FROM AUTHOR]