An Introduction to Galois Theory

This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics.

Groups of Sets

The binary operation * for combining sets is defined as the union of two sets minus their intersection. Prove the set of all subsets of a set S together with the binary operation * forms a group.

Shuffles

An environment for exploring the properties of small groups.

What's a Group?

Stage: 5 Challenge Level:

A group is a set of elements together with a binary operation (which we denote here by $*$) with the following properties:

1. CLOSURE For all elements $a$ and $b$ in the group, the element $a*b$ is also in the group.
2. ASSOCIATIVITY If $a, b$ and $c$ are in the group then $(a*b)*c = a*(b*c)$.
3. IDENTITY The group contains an element $e$, called the identity, such that if $a$ is in the group then $a*e = e*a = a$.
4. INVERSES If $a$ is an element in the group then there is also an element in the group $a'$, called the inverse of $a$, such that $a*a' = a'*a = e$.

Some groups, which are called COMMUTATIVE or ABELIAN, have the property that, for all pairs of elements in the group, $a*b=b*a$.