Abstract Algebra Study Guide

1. Operations

Closures, associativity, commutative, identity elements, inverse elements

2. Groups

Definition of group, modular groups Z_n, symmetric groups S_n, order of a group

3. Subgroups

definition of subgroup, test for being a subgroup, cyclic groups, generators, order of an element, Lagrange’s theorem

4. Mappings

one-to-one, onto, homomorphisms, isomorphisms, kernel

5. Quotient Groups

Normal subgroups, cosets, Fundamental Homomorphism Theorem

6. Rings

Definitions of ring, ring with unity, zero divisor, integral domain, and field