### abstract algebra

An automorphism is an isomorphism between a group and itself. The identity
function (x -> x) is always an isomorphism, which we consider trivial. Use Sage
to construct a nontrivial automorphism of the cyclic group of order 12. Check that
the mapping is both onto and one-to-one by computing the image and kernel and
performing the proper tests on these subgroups. Now construct all of the possible
automorphisms of the cyclic group of order 12.