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A subgroup H of a group G is called an ICΦs -subgroup of G if (H ∩ [ H , G ]) H G / H G ≤ Φ (H / H G) H s G / H G , where H G = ∩ g ∈ G H g and Φ (H / H G) is the Frattini subgroup of H / H G and HsG is the largest s-permutable subgroup of G contained in H. In this paper, we investigate the structure of finite group G under the assumption that some subgroups of G are ICΦs -subgroups of G, some results are obtained. [ABSTRACT FROM AUTHOR]

LetGbe a finite group, andT(G) be the sum of all complex irreducible character degrees ofG. In this paper, we get the exact lower bound of |G|∕T(G) for a non-r-solvable groupG. [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]

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or As if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s. [ABSTRACT FROM AUTHOR]