Group Theory

Definition

Group theory is a branch of abstract algebra that studies groups: sets equipped with an operation that satisfies certain algebraic rules.

Group theory investigates:

• the structure and properties of groups;
• subgroups and normal subgroups;
• group homomorphisms and isomorphisms;
• quotient groups;
• symmetry and transformations.

For example, the set of integers Z under addition forms a group:

(Z, +).

Group theory is used to study symmetry in mathematics, geometry, physics, chemistry, cryptography, and many other fields.