What is a Group?
The Core Object in Abstract Algebra
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.
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.
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.
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.
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.
Mathematica and the Wolfram Language
“Mathematics is the art of giving the same name to different things.”
Henri Poincaré