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Let G be a finite group and H ≤ G. The authors study the structure of finite groups G having a subgroup H which is maximal in 〈H, Hg〉 for some g ∈ G. Some results on the structure of 〈H, Hg〉 and G are set up. Especially, a generalization of Baer′s theorem is established.

In this paper, we study how certain arithmetical conditions on the lengths of conjugacy classes of primary elements and π-elements of a finite group G influence the structure of G, some new results are obtained and several well-known results in the literature are developed. In particular, the structure of a kind of outer p-supersolvable groups is described, some new conditions implying π-supersolvability are found, and the upper bound for the order of Sylow subgroups is discussed. [ABSTRACT FROM AUTHOR]

We introduce a new subgroup embedding property of finite groups called CSQ-normality of subgroups. Using this subgroup property, we determine the structure of finite groups with some CSQ-normal subgroups of Sylow subgroups. As an application of our results, some recent results are generalized.

Let Φ a (G) and Φ n (G) denote the intersection of all abnormal and normal maximal subgroups of G, respectively. S is called Φ a (or Φ n )-supplemented in G if there exists a proper normal subgroup N of G such that G = SN and S ∩ N ≤ Φ a (S) (or Φ n (S) ). In this paper we characterize many properties of groups by using Φ a and Φ n -supplementation. Communicated by Sudarshan Kumar Sehgal [ABSTRACT FROM AUTHOR]

Let G be a finite group, p a prime, P a Sylow p-subgroup of G and d a power of p such that 1 < d < | P |. Let G p * denote the unique smallest normal subgroup of G for which the corresponding factor group is abelian of exponent dividing p – 1. Let F 1 , F 2 , F 3 be classes of all p-groups, p-nilpotent groups and p-supersolvable groups, respectively, G F be the F -residual of G. Let E ∈ { G F 1 , (G p *) F 1 , G F 2 , G F 3 }. We prove that if H ∩ E is weakly s-semipermutable in G for all normal subgroups H of P with | H | = d , then either G ∈ F 3 and G ∈ F 2 if p is the least prime factor of | G | , or else | P ∩ E | > d ; and for all subgroups H of P with | H | = d and 4 (if p = 2), then G ∈ F 3 and G ∈ F 2 if p is the least prime factor of | G |. Many known theorems become Corollaries of our results. Communicated by Peter Sin [ABSTRACT FROM AUTHOR]

Let G be a finite p-group. If |CG(x): ⟨ x ⟩ | ≤pkfor every noncentral element x of G, then G is called a k − 𝒩𝒞 −p-group. In this article, we give a classification of k − 𝒩𝒞 −p-groups, where k = 1, 2 and p > 2. [ABSTRACT FROM PUBLISHER]

Let G be a finite group and p a prime. In this article, we give a criterion of p-nilpotency of G by means of the normalizer and a generalization of normality of subgroups. [ABSTRACT FROM PUBLISHER]

In this article, we classify p-groups whose non-normal subgroups have index p in their normalizers. This answers a problem posed by Berkovich and Janko. [ABSTRACT FROM AUTHOR]