MATH411

Describe examples of groups, rings, and integral domains with various combinations of properties.

Describe mathematical structures that serve as counterexamples to supposed assertions in group theory and ring theory.

Use concepts and notation of the course in an abstract sense appropriate to their definitions rather than relying on preconceived notions.

Perform calculations with cyclic groups, modular arithmetic, permutation groups, cosets, factor groups, orders of elements, and the Fundamental Theorem of Finite Abelian Groups.

Determine whether a set is subgroup of a group, a normal subgroup of a group, or a subring of a ring.

State famous and 'named' theorems of group theory and ring theory (such as the First Isomorphism Theorem).

Apply known results, concepts, and techniques of group theory and ring theory to investigate new situations and prove other results.

Reason mathematically and correctly.