Problem 1: Let $A,B,C, D$ be $n$ by $n$ matrices. Let $ABCD = I$, the identity matrix. Prove that $ABCD = BCDA = CDAB = DABC$.
Problem 2: Show that every group with an infinite number of elements has an infinite number of subgroups.
Problem 3: Find all groups with 5 or less elements up to isomorphism.
Problem 4: Find all rings with 5 or less elements up to isomorphism.
Problem 5: Find all fields with 5 or less elements up to isomorphism.
Problem 6: If the binary operation of the group $G$ is addition, and $1 \in G$, find the set of numbers that must be in $G$.
Problem 7: Show that a set of all invertible matrices of a certain size along with matrix multiplication is a group.
Problem 8: Let $n$ be an integer greater than 1. Prove or disprove that the integers modulo n along with modular addition and modular multiplication is a ring.
Problem 9: Come up with an infinite number of groups that aren't isomorphic to each other.
Problem 10: Come up with an infinite number of rings that aren't isomorphic to each other.
Problem 11: Come up with an infinite number of fields that aren't isomorphic to each other.
Problem 12: Come up with an infinite number of binary operations that are defined on the integers, surjective on the integers, and associative.
Problem 13: Find all subsets of the rational numbers that when combined with multiplication are groups isomorphic to the group of integer addition.
Problem 14: Show that any cyclic group with an infinite order is isomorphic to the group of integer addition.
Problem 15: Find all subsets of the rational numbers $\mathbb{Q}$ such that when combined with multiplication constitute a field.