# Chapter 4 Group theory

A group is a very simple mathematical object consisting of a set and a way of combining two elements of the set to produce another, called the group operation. This group operation has to obey three rules mimicing those obeyed by the symmetries of a physical object called the group axioms. In this section we look at a wide variety of examples of groups and explore some of their basic properties.

Learning objectives for this section:

• Be able to state the group axioms and to verify whether a given set and binary operation form a group.
• Define subgroup, identity element, inverse, associativity, order of an element, order of a group, group table, inverse, cyclic group, abelian/commutative group, Cartesian product of groups.
• Compute orders, powers, and inverses in concrete examples.
• Determine whether a group is cyclic or abelian.
• Determine whether a given subset is a subgroup, including using Lagrange’s theorem.
• Be able to define and compute with cyclic groups, the additive group mod n, the multiplicative group mod p, the symmetric group, the dihedral group
• State Lagrange’s Theorem and Fermat’s Little Theorem.