On Non-commuting Sets in Certain Finite p-Groups

Let G be a group. A subset X of G is said to be non-commuting if xy not equal yx for any x, y is an element of X with x not equal y. If vertical bar X vertical bar > vertical bar Y vertical bar for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, the bound for the cardinality of a maximal non-commuting set in a finite p-group G is determined, where G is a non-abelian p-group given by a central extension as 1 -> Z(p)(m) -> G -> Z(p) X ... X Z(p) -> 1 and its derived subgroup has order p.