TAILIEUCHUNG - Generalized hypercenter of a finite group

Let G be a finite group. In this paper, we introduce the concept of super generalized supersolvably embedded subgroup of a group G and give a new characterization of the generalized hypercenter of G. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 658 – 663 ¨ ITAK ˙ c TUB ⃝ doi: Generalized hypercenter of a finite group Mohamed Ezzat MOHAMED1,∗, Mohammed Mosa AL-SHOMRANI2 1 Faculty of Arts and Science, Northern Borders University, Rafha, Saudi Arabia 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Received: • Accepted: • Published Online: • Printed: Abstract: Let G be a finite group. In this paper, we introduce the concept of super generalized supersolvably embedded subgroup of a group G and give a new characterization of the generalized hypercenter of G . Key words: S -quasinormally embedded, generalized supersolvably embedded, super generalized supersolvably embedded, supersolvable 1. Introduction All groups considered in this paper will be finite and G always means a finite group. We use conventional notions and notations, as in Doerk and Hawkes [4]. Two subgroups H and K of a group G are said to permute if HK = KH . Thus, we have that H and K permute if and only if HK is a subgroup of G . We say, following Kegel [5], that a subgroup of a group G is S -quasinormal in G if it permutes with every Sylow subgroup of G . In 1998, Ballester-Bolinches and Pedraza-Aguilera [3] introduced the following definition: a subgroup H of a group G is said to be S -quasinormally embedded in G if each Sylow subgroup of H is a Sylow subgroup of some S -quasinormal subgroup of G . Obviously, every S -quasinormal subgroup is S -quasinormally embedded. The converse does not hold in general. The Sylow 2-subgroups of S3 , the symmetric group of degree 3 , are S -quasinormally embedded but not S -quasinormal subgroups of S3 . Agrawal [1] defined the generalized center genz(G)of a group G to be the subgroup is S -quasinormal in G > . The generalized hypercenter genz∞ (G), is the largest term of the .

• Toán học
• S quasinormally embedded
• Generalized supersolvably embedded
• Super generalized
• Supersolvably embedded
• Generalized hypercenter