Engel‐4 Groups of Exponent 5

We show that if G is a group of exponent 5, and if G satisfies the Engel‐4 identity [ x,y,y,y,y ]=1, then G is locally finite. By a result of Traustason, this implies that Engel‐4 5‐groups are locally finite. We also show that a group of exponent 5 is locally finite if and only if it satsifies the identity [ x , [ y , z , z , z , z ] , [ y , z , z , z , z ] ] = 1.This result implies that a group of exponent 5 is locally finite if its three generator subgroups are finite. 1991 Mathematics Subject Classification : 20D15, 20F45, 20F50.