We find the smallest field containing the number 1 when using the usual addition and multiplication of real numbers.
We show, step-by-step, that the set of all 2 x 2 matrices with real entries forms a ring under addition and multiplication.
We show that every cyclic group is isomorphic to either the set of integers or the set of integers mod some integer under addition.
We show that the composition of two isomorphisms is again an isomorphism.
We show that the image of an Abelian group under a homomorphism is itself an Abelian group. That is, homomorphisms preserve commutativity.