On the Schur multiplier of p-groups

The referee for [(4) under bar] has drawn my attention to a useful result of Gaschutz, Neubuser and Yen [(5) under bar] on the order of the Schur multiplier M(G) of a finite p-group. As a means of advertising the potential of this result and its generalisations given in [(4) under bar], we use them to obtain a short proof of the following theorem. Parts (i) and (ii) of the theorem are due to Berkovich [(1) under bar], part (iii) is due to Zhou [(6) under bar], and part (iv) is new. Our method is quite different to that of [(1) under bar] and [(6) under bar], and can be used with the results in [(4) under bar] to obtain further information for t greater than or equal to 4.