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Let G be a group. A subset S of G is said to normally generate G if G is the normal closure of S in G: In this case, any element of G can be written as a product of conjugates of elements of S and their inverses. If g ∈ G and S is a normally generating subset of G, then we write ||g||S for the length of a shortest word in ConjG(S±1):= {h-1sh|h ∈ G, s ∈ S or s-1 ∈ S} needed to express g: For any normally generating subset S of G; we write ||G||S = sup{||g||S | g ∈ G}. Moreover, we write Δ(G) for the supremum of all ||G||S, where S is a finite normally generating subset of G; and we call Δ(G) the conjugacy diameter of G. In this paper, we derive the conjugacy diameters of the semidihedral 2-groups, the generalized quaternion groups and the modular p-groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups. [ABSTRACT FROM AUTHOR]

Abstract–Let p be a prime number, G be a p-solvable finite group and P be a Sylow p-subgroup of G. We prove that G is p-supersolvable if N G (P) is p-supersolvable and if there is a subgroup H of P with P ′ ≤ H ≤ Φ (P) such that H is s-semipermutable in G. As applications, we simplify the proofs of some known results and also generalize some known results. [ABSTRACT FROM AUTHOR]