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In this paper we determine the nonabelian tensor square of the free 2-Engel group of rank 3 and of the Burnside group on 3 generators of exponent 3. Both tensor squares are nilpotent groups of class 2. The calculatory method used is based on the concept of a crossed pairing. Some of the expansion formulas and verifications occuring in this context require extensive calculations. A computer program written in the GAP language assisted in completing these symbolic computations.

The nonabelian tensor square G⊗G of a group G is generated by the symbols g⊗h, g, h ∈ G, subject to the relations,for all g, g′, h, h′ ∈ G, where The tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J. L. Loday in [4] and [5], extending ideas of Whitehead in [6].