Group Theory Explained in 5 Minutes

Group Theory Explained in 5 Minutes

Group Theory is a huge branch of mathematics and has applications everywhere, including physics. So what's a group?

Suppose you have a set of mathematical objects S={a,b,c,..}

What type of objects? Any type, it doesn't matter.

Now suppose you also have an operation * with the following properties..

1. It operates on any two objects in S to produce another object in S. In other words, if you take two objects in S, say a and b, then a*b is also an object in S. Mathematicians say that S is "closed" under the operation.

2. It's associative. This just means that for any 3 objects in S, say a, b, c then a*(b*c)=(a*b)*c. In other words, providing we keep the order of the objects the same we can "associate" them in any way we want. This may seem trivial, but remember, the operation * is more sophisticated than the ordinary multiplication we use with numbers.

The operation * is called a binary operation for the simple reason that it operates on two objects.

To make S into a Group all we need are two more rules.

1. There exists a special object in S, let's call it e, such that e*a=a for every a in S. This object e is called the identity, and it's easy to prove there can only be one identity object in a Group. So this is a unique object.

2. Every object a in S has a partner, let's call it a', such that a*a'=e. This partner is called the inverse of a.

That's it. S is now a Group. A Group is one of the most fruitful concepts in all of mathematics. But from the above definition you would never guess!

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Content written and posted by Ken Abbott
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