Third Engel groups

Abstract

We present some new results on third Engel groups which are motivated by computer calculations but are not dependent on them. They include:• for n > 2 every n-generator third Engel group is nilpotent of class at most 2n – 1;• the fifth term of the lower central series of a third Engel group has exponent dividing 20;• the subgroup generated by fifth powers of elements in a third Engel group is nearly centre-by-metabeliami;and a normal form theorem for freest third Engel groups without elements of order 2.