p -GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY p -GROUPS.

Let G be a p -group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p -groups. [ABSTRACT FROM AUTHOR]