On the nonabelian tensor square of a 2-Engel group

Abstract

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.

On the nilpotency class and solvability length of nonabelian tensor products of groups

“…These bounds generalize those for the nonabelian tensor square which appear in [2] and [3], respectively. …”

“…Thus we have cl G G cl G H and l G G l G H , if G is nilpotent of class two. On the other hand, it was shown in [2] that cl e e 2, where e is the free 2-Engel group of rank three. Observing that cl e H l e H 1, we have cl e e cl e H 1 and l e e l e H 1.…”

“…All that needs to be shown is that the crossed pairing properties hold for D. Towards that end, suppose gY g H P G and h P H. For the example of this section we use Proposition 4.3 with Q e, the free three generator, two-Engel group, and Proposition 4.4 with G F, the crossed pairing e  e 3 L, given in Theorem 3.6 of [2]. It should be noted that [2], in particular Proposition 3.3 and Theorem 3.6, is necessary for a complete understanding of the following example. E x a m p l e 4 .…”

“…We now construct the homomorphisms embedding G and H in e. By Proposition 3.3 in [2], every e P e has a unique presentation as e a a b b c g aY b & aY c s bY c t aY bY c l , where aY bY gY &Y sY t P Z and l P Z 3 . Next let g s 1 a 1 s 2 a2 s 3 a3 s 4 a4 s 1 Y s 2 a 5 s 4 Y s 1 a6 P G and define q X G 3 e by qg a a 1 b a2 aY b a 5 aY c a3 bY c a4 aY bY c a6 .…”

On Nonabelian Tensor Analogues of 2-Engel Conditions

“…It follows directly from Proposition 4 that G ⊗ G is abelian. From the same proposition we obtain ([x, y, z In contrast with this result, there exists a 2-Engel group G such that cl (G ⊗ G) = 2 [2]. The following is a tensor analogue of Proposition 3.…”