Abelian Covers and Non-Commuting Sets in a Non-Abelian p-Group Which its Central Quotient is Metacyclic

Let G be a group. A set S in G is said to be non-commuting if xy≠yx for any two distinct elements x,y∈S. We define w(G) to be the maximum possible cardinality of a non-commuting set in G. In this paper, we determine w(G) for a finite non-abelian p-group G such that G/Z(G) is metacyclic by obtaining an abelian centralizers cover of this group. As a consequence, we show that the set of all commuting automorphisms of a finite non-abelian p-group G such that G/Z(G) is metacyclic, forms a subgroup of Aut(G).