Group Axioms

A group is a set \(G\) of elements with an operation \(*\) that let's you "combine" two elements. They satisfy these properties:

  • CLOSURE: For any two elements \(a, b\in G\), their "product" \(a*b\) is also in \(G\). Mathematicians say that \(``G\) is closed under the operation \(* "\)

  • IDENTITY: There's an element \(e\) that acts like the numbers \(0\) for addition and \(1\) for multiplication. That is, \(a*e=a\) and \(e*a=a\). Combining the identity element with any other element using \(*\) has no effect.

  • INVERSES: Every element \(g\in G\) has an inverse. This is an element which we write as \(g^{-1}\) that when multiplied by \(g\) give you the identity \(e\). To be super explicit, \(g*g^{-1}=e\) and \(g^{-1}*g=e\).

  • ASSOCIATIVITY: The associative property from arithmetic also applies to elements in the group: \(a*(b*c)=(a*b)*c\). This is key so you can do things like cancel terms in an equation and other common algebraic operations.

Applications of Groups

How these abstract objects are used concretely

  • Cryptography

    Many of the algorithms that keep digital communications secure rely on groups in their execution.

  • Quantum Mechanics

    In theoretical physics, group theory is an effective tool at describing many phenomena.

  • Games & Theory

    Many puzzles, like the Rubik's Cube, can be described and studied using the results from group theory.

FAQ about Groups

  • Why create and study such an abstract idea?

    Mathematicians noticed that the same discoveries were being made over and over. By creating a general - and useful - concept, the group lets you solve the problems once, and then apply the solution to specific applications.

  • Are groups useful in any way?

    Very much yes! As is often the case in mathematics, it's easier to talk about things "in abstract" rather than teach you everything you need to know to see how they are used in other fields. But groups are definitely used in physics and computer science.

  • Are groups completely understood?

    Very much no! While a lot of progress has been made, it's only the tip of the iceberg. For example, we have a list of the building blocks for finite groups (simple groups), but we don't yet know all the ways you can combine them to make larger, more complicated groups.

  • How many groups are there?

    There is an infinite variety of groups. Some of them are simple and familiar, like the integers under addition or modular arithmetic, while others are extremely difficult to simply define.

Computational Group Theory

Mathematica and the Wolfram Language

Mathematica has built-in tools that let you programmatically build and explore groups. This includes abelian groups, symmetric groups, and the sporadic simple groups - including the Monster group!
Learn More

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“Mathematics is the art of giving the same name to different things.”

Henri Poincaré