## Field Generated by 1

We find the smallest field containing the number 1 when using the usual addition and multiplication of real numbers.

## A Ring of Matrices

We show, step-by-step, that the set of all 2 x 2 matrices with real entries forms a ring under addition and multiplication.

## Representations of Cyclic Groups

We show that every cyclic group is isomorphic to either the set of integers or the set of integers mod some integer under addition.

## Compositions of Isomorphisms

We show that the composition of two isomorphisms is again an isomorphism.

## Homomorphisms Preserve Commutativity

We show that the image of an Abelian group under a homomorphism is itself an Abelian group. That is, homomorphisms preserve commutativity.