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.
I take some time away from doing algebra to look at an upcoming video game, New World, and discuss its potential.
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.