Left 3-Engel elements in groups of exponent 5

It is still an open question whether a left 3-Engel element of a group G is always contained in the Hirsch–Plotkin radical of G. In this paper we begin a systematic study of this problem. The problem is first rephrased as saying that a certain type of groups are locally nilpotent. We refer to these groups as sandwich groups as they can be seen as the analogs of sandwich algebras in the context of Lie algebras. We show that any 3-generator sandwich group is nilpotent and obtain a power-conjugation presentation for the free 3-generator sandwich group. As an application we show that the left 3-Engel elements in any group G of exponent 5 are in the Hirsch–Plotkin radical of G.