A test element in a group G is
an element g with the property that if f(g) = g for an endomorphism f of G to G
then f must be an automorphism. A test element in a free group is called a testword. Nielsen gave the first example of a test word by showing that in the free group
on x,y the commutator [x,y] satisfies this property. T. Turner recently characterized
test words as those elements of a free group contained in no proper retract. Since free
factors are retracts, test words are therefore very strong forms of non-primitive
elements. In this paper we give some new examples of test words and examine the
relationship between test elements and several other concepts, in particular
generic elements and almost-primitive elements (APE’s). In particular we
show that an almost primitive element which lies in a certain type of verbal
subgroup must be a test word. Further using a theorem of Rosenberger on
equations in free products we prove a result on APE’s, generic elements and
test words in certain free products of free groups. Finally we examine test
elements in non-free groups and introduce the concept of the test rank of a
group.
