LAUSR

Abstract Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism φ : G → G , φ ( g ) = g implies that φ is an automorphism. Let F ( X ) be a free group on a finite non-empty set X, and let X = X 1 ∐ X 2 ∐ … ∐ X r be a finite partition of X into r ≥ 2 non-empty subsets. For i = 1 , 2 , … , r , let u i ∈ 〈 X i 〉 ≤ F ( X ) , and let w ( z 1 , … , z r ) be a word in the variables z 1 , … , z r . We give several sufficient conditions on u i ( 1 ≤ i ≤ r ) and w for w ( u 1 , … , u r ) to be a test element of F ( X ) . As an application of these results, we give examples of test elements of a free group of rank greater than two that are not test elements in any pro-p completion of the group.