On Generalized Hypercenter of a Finite Group

On the 𝔉‐hypercentre of a finite group

“…Recently, Asaad and Ezzat in [2] describe the generalized hypercentre genz ∞ (G) in terms of the following embedding property:…”

“…Asaad and Ezzat proved that genz ∞ (G) is the product of all GSE normal subgroups of G (see [2,Theorem 3.5]). …”

Finite Groups Whose Generalized Hypercenter Contains Some Subgroups of Prime Power Order

“…Recently, Asaad and Ezzat Mohamed [3] give a new characterization of genz G by introducing the following definition: A normal subgroup H of a group G is a generalized supersolvably embedded (GSE) in G if there exists a chain 1 = H 0 ≤ H 1 ≤ · ≤ H n = H such that H i is S-quasinormal in G and H i+1 H i is a prime, where 0 ≤ i ≤ n − 1. It is easily verified that, if H and K are normal GSE subgroups of G, then HK is GSE in G. From this, a group G has a unique maximal generalized supersolvably embedded subgroup of G, and it is denoted by GSE G .…”

Generalized hypercenter of a finite group

“…where genz i+1 (G)/genz i (G) = genz(G/genz i (G)) for all i > 0. Asaad and Ezzat Mohamed [2] gave a new characterization of genz ∞ (G) by introducing the following definition: a normal subgroup H of a group G is a generalized supersolvably embedded ( GSE)in G if there exists a chain…”