On the triple tensor product of prime-power groups

Abstract

Abstract Let G be a finite p-group, and let ⊗ 3 G {\otimes^{3}G} be its triple tensor product. In this paper, we obtain an upper bound for the order of ⊗ 3 G {\otimes^{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, 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 ⊗ 3 G {\otimes^{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, 10, 4219–4234], we show that, when G is a nilpotent group of class at most 4, exp ( ⊗ 3 G ) {\exp(\otimes^{3}G)} divides exp ⁡ ( G ) {\exp(G)} .