A note on pronormal p -subgroups of finite groups
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A note on pronormal p-subgroups of finite groups Suli Liu1 · Haoran Yu1 Received: 9 September 2020 / Accepted: 3 November 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this note, we investigate the influence of pronormal p-subgroups on the structure of finite groups. We not only simplify, but also generalize some main results of Liu et al. (J Algebra Appl 19:2050110, 2020), Shen et al. (Monatshefte Math 175:629–638, 2014) and Yu and Lai (Monatshefte Math 181:745–747, 2016). We also prove that for p-subgroups of finite groups, the concept of pronormal subgroups, weakly normal subgroups and weakly closed subgroups are equivalent. Keywords p-subgroups · Pronormal subgroups · Weakly closed subgroups Mathematics Subject Classification 20D10
1 Introduction Throughout the paper, we suppose G is a finite group and p is a prime. In his lectures in Cambridge, P. Hall introduced the concept of pronormal subgroups of finite groups. Definition 1.1 ([3], Page 241) A subgroup H of a group G is said to be pronormal in G if for each g ∈ G, the subgroups H and H g are conjugate in H , H g . It is easy to see that normal subgroups and Sylow subgroups are pronormal subgroups. Pronormality is one of the most significant properties pertaining to subgroups of finite groups, so it has been widely studied, for example, see [1,4,8,9]. Recently, Liu et al. [6] investigated the structure of finite groups whose subgroups of prime power order are pronormal. One of their main theorems is as follows.
Communicated by John S. Wilson.
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Haoran Yu [email protected] Suli Liu [email protected]
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School of Mathematics, Jilin University, Changchun 130012, China
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S. Liu , H. Yu
Theorem 1.2 ([6], Theorem 1.2) Let p be the smallest prime dividing the order of a group G and let P be a Sylow p-subgroup of G. If every maximal subgroup of P is pronormal in N G (P) and P G, then G is p-nilpotent. In this short note, we not only simplify, but also generalize Theorem 1.2 as follows. Theorem 1.3 Let p be a prime dividing the order of a group G with (|G|, p − 1) = 1 and let P be a Sylow p-subgroup of G. If P has a subgroup D such that 1 < |D| < |P| and all subgroups of P with order |D| and all cyclic subgroups of P with order 4 (if P is a non-abelian 2-group and |D| = 2) are pronormal in N G (P) and P G, then G is p-nilpotent. In Sect. 4, we will show that for p-subgroups of finite groups, the concept of pronormal subgroups, weakly normal subgroups and weakly closed subgroups are equivalent. Hence Theorem 1.3 also simplify and generalize [[10], Theorem 3.9] and [[12], Theorem 3.1].
2 Preliminaries Lemma 2.1 Let p be a prime dividing the order of a group G, P be a Sylow p-subgroup of G, and H be a subgroup of P. If H is pronormal in N G (P), then H N G (P). Proof Assume that there exists g ∈ N G (P) such that H = H g , and we work to obtain a contradiction. Since g ∈ N G (P) and H ≤ P, it follows that H g ≤ P g = P. Let L = H , H g . By the definition of pronormal subgroups, there exists k ∈ L
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