On p -CAP-subgroups of finite groups
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On p-CAP-subgroups of finite groups Zhichao GAO1 , Shouhong QIAO2 ,
Huaguo SHI3 , Long MIAO1
1 School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China 2 School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China 3 Education faculty, Sichuan Vocational and Technical College, Suining 629000, China
c Higher Education Press 2020
Abstract Suppose that G is a finite group and H is a subgroup of G. H is said to be a p-CAP-subgroup of G if H either covers or avoids each pd-chief factor of G. We give some characterizations for a group G to be p-solvable under the assumption that some subgroups of G are p-CAP-subgroups of G. Keywords Maximal subgroups, 2-maximal subgroups, p-CAP-subgroups MSC2020 20D10, 20D20 1
Introduction
All groups considered in this paper will be finite. We shall adhere to the notation employed in [3,8]. Denoted by |G| the order of a group G, π(G) denotes the set of primes dividing |G|, and HG denotes the core of H in G. We write A l G to mean that A is a maximal subgroup of G. We use A : B to denote a split extension of a group A by another group B. Write T = {H | ∃ M l G s.t. H l M } to mean that T is the set of second maximal subgroup of G. And write T1 = {H | ∃ M l G s.t. H l M, HG = MG }. Clearly, T1 ⊆ T. Let G be a group and H/K a chief factor of G. We say that a subgroup A covers H/K if HA = KA; or A avoids H/K if H ∩ A = K ∩ A. We say that A has the cover and avoidance properties in G if A either covers and avoids every chief factor of G, A is called a CAP-subgroup for short. The cover and avoidance property of subgroups was first studied by Gasch¨ utz [6] to characterize the solvable groups, later by Gillam [2] and Tomkinson [11]. Received June 12, 2020; accepted September 25, 2020 Corresponding author: Long MIAO, E-mail: [email protected]
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Ezquerro [4] gave some characterizations for a group G to be p-supersolvable and supersolvable under the assumption that all maximal subgroups of some Sylow subgroups of G have the cover and avoidance property in G. Guo and Shum [7] pushed further this approach and obtained some characterizations for a solvable group and a p-solvable group based on the assumption that some subgroups have the cover and avoidance property. He and Wang [9] introduced the p-cover and avoidance property, which is a generalization of the cover and avoidance property. We say that a subgroup A has the p-cover and avoidance property in a group G if A either covers or avoids each pd-chief factor of G, A is called a p-CAP-subgroup for short. Recall that a chief factor H/K is said to be a pd-chief factor if p divides |H/K|. In this paper, we will generalize two results of Guo and Shum [7]. Also, we will give some characterizations for a group G to be p-solvable under the assumption that some subgroups of G are p-CAP-subgroups.
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Preliminaries
Below, we list some basic results which will be used in the sequel. Lemma 2.1 [9, Lemma 2.5] Let H be a p-CAP-subgroup of G and N E G. Then the following statement
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