Define and use Subgroups, Ideals, Homomorphisms and other basic concepts about Groups and Rings,
Analyze and produce proofs of statements involving Groups and Rings,
Use some fundamental examples such as the Symmetric Group and its Subgroups, Cyclic Groups, Matrix Groups, and Polynomial Rings,
Distinguish basic Algebraic structures from each other up to Isomorphism,
Use the three Isomorphism theorems for Groups and Rings,
Use Direct and Semi-direct Product Constructions,
Use Group Actions to study structures of Groups and to do effective counting in Groups,
Apply Sylow Theorem and the Structure Theorem on Finitely Generated Abelian Groups,
State basic Ring types such as Local Rings, Principal Ideal Domains, Unique Factorization Domains.