Conjugate p-elements of Full Support that Generate the Wreath Product Cp wr Cp

For a symmetric group G:=Sym(n) and a conjugacy class X of involutions in G, it is known that if the class of involutions does not have a unique fixed point, then - with a few small exceptions - given two elements a,x in X, either is isomorphic to the dihedral group D8, or there is a further element y in X such that and are both isomorphic to D8 (P. Rowley and D. Ward, On pi-Product Involution Graphs in Symmetric Groups. MIMS ePrint, 2014). One natural generalisation of this to p-elements is to consider when two conjugate p-elements generate a wreath product of two cyclic groups of order p. In this paper we give a necessary and sufficient condition for this in the case that our p-elements have full support