A test element g in G is an element with the property that the onlyendomorphisms that fix g are automorphisms. This provides a means of recognizing automorphisms that requires only one piece of information.
Free groups have many test elements (test words in this context) and it is possible to describe them completely and to algorithmically decide whether a given word isa test word. The fact that automorphic equivalence in a free group is efficientlydecidable means that test words provide an effective method of recognizingautomorphisms in a free group (with minimal information).
There are interesting results in other classes of groups as well -- for examplefinitely generated abelian groups. The class of hyperbolic groups generalizes that of free groups. I will describe progress on the following conjecture.
Conjecture: All torsion free hyperbolic groups have test elements.