On the triple tensor product of prime-power groups

Let G be a finite p-group, and let circle times(3)G be its triple tensor product. In this paper, we obtain an upper bound for the order of circle times(3)G, which sharpens the bound given by G. Ellis and A. McDermott, [Tensor products of prime-power groups, J. Pure Appl. Algebra 132 (1998), no. 2, 119-128]. In particular, when G has a derived subgroup of order at most p, we classify those groups G for which the bound is attained. Furthermore, by improvement of a result about the exponent of circle times(3)G determined by G. Ellis [On the relation between upper central quotients and lower central series of a group, Trans. Amer. Math. Soc. 353 (2001), no. 10, 4219-4234], we show that, when G is a nilpotent group of class at most 4, exp(circle times(3)G) divides exp(G).