Finite groups whose n-maximal subgroups are σ-subnormal

Abstract

Let σ = {σ i |i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member = 1 of H is a Hall σ i -subgroup of G, for some i ∈ I, and H contains exact one Hall σ i -subgroup of G for every σ i ∈ σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set set H such that HAwhere M i is a maximal subgroup of M i−1 , i = 1, 2, . . . , n, then M is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal (σ-quasinormal, respectively) in G but, in the case n > 1, some (n − 1)maximal subgroup is not σ-subnormal (not σ-quasinormal, respectively)) in G, we write m σ (G) = n (m σq (G) = n, respectively).In this paper, we show that the parameters m σ (G) and m σq (G) make possible to bound the σ-nilpotent length l σ (G) (see below the definitions of the terms employed), the rank r(G) and the number |π(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when m σ (G) = |π(G)|. Some known results are generalized.

ON THEσ-NILPOTENT NORM AND THEσ-NILPOTENT LENGTH OF A FINITE GROUP

“…If E is a π -group, it is clear. Now suppose that E is not a π -group and let x be an element of E of odd prime order p. Then, x ∈ N σ (G) ∩ E ≤ N σ (E) by Proposition 2.5 (3). It is clear also that E is σ -soluble.…”

“…By the σ -nilpotent length (denoted by l σ (G) [3]) of a σ -soluble group G, we mean the length of the shortest normal chain of G with σ -nilpotent factors. REMARK 1.1.…”

On an open problem of Guo-Skiba

“…Let A and B be subgroups of G. Following [5], we say that A forms an irreducible pair with B if AB = BA and A is a maximal subgroup of AB. (2) If M is a maximal subgroup of a σ-soluble group G, then |G : M| is σ-primary.…”

“…If each n-maximal subgroup of G is σ-subnormal in G but, in the case n > 1, some (n − 1)maximal subgroup is not σ-subnormal in G, then we write m σ (G) = n (see [5]). If G is a soluble group, the rank r(G) of G is the maximal integer k such that G has a G-chief factor of order p k for some prime p (see [6, p. 685]).…”

On -Quasinormal Subgroups of Finite Groups

“…A chief factor H/K of G is said to be σ-central in G (as defined in [10]) if (H/K) (G/C G (H/K)) is σ-primary; G is called σ-nilpotent [10] if every chief factor of G is σ-central. In view of [3,Proposition 2.7], G is σ-nilpotent if and only if G = G 1 × • • • × G t for some σ-primary groups G 1 , . .…”