Groups that are Pairwise Nilpotent

Abstract

In this paper we study groups generated by a set X with the property that every two elements in X generate a nilpotent subgroup.

Groups generated by a finite Engel set

“…For all x, y ∈ S, the subgroup x, y is nilpotent by Theorem 3.5. Thus the claim follows by Proposition 1 of [3].…”

“…For all x, y ∈ S, the subgroup x, y is nilpotent by Theorem 3.5. Thus the claim follows by Proposition 1 of [3]. Using Theorem 3.5, we now present a criterion for nilpotency of a finite soluble group depending on information on its Sylow subgroups.…”

“…However, a group generated by a finite Engel set is not necessarily nilpotent: Golod's examples show that there exist infinite non-nilpotent groups generated by an Engel set with three or more elements (see [5]). Furthermore, if S is an Engel set of size three, then an easier example of a non-nilpotent group generated by S is the wreath product of the alternating group of degree 5 with the cyclic group of order 3: it has a presentation of type (r, s, t) (see [3]), i.e. S = {a, b, c} where a, b is nilpotent of class r, a, c is nilpotent of class s and b, c is nilpotent of class t. All these groups are not soluble, but the nilpotency does not hold even in the soluble case.…”

“…S = {a, b, c} where a, b is nilpotent of class r, a, c is nilpotent of class s and b, c is nilpotent of class t. All these groups are not soluble, but the nilpotency does not hold even in the soluble case. In [3] it was shown that every group with a presentation of type (1,2,2) is soluble of length at most 3 and that there are non-nilpotent groups of this type.…”

“…In fact, we construct by GAP (see [4]) a non-nilpotent counterexample which is abelian-by-(nilpotent of class 3). On the other hand, some of the counterexamples in [3], mentioned above, are abelian-by-(nilpotent of class 2) and generated by an Engel set of size three.…”