The concept of a “Group” is fundamental to modern mathematics. It’s a rather simple concept, but it’s implications and power are truly profound.
Definition of a Group
A set , with the operation , is a group if the operation satisfies the following:
- is associative;
- There exists an element such that and for all elements ;
- For all , there exists an such that and .
We can denote the group and the operation together with this notation: .
Examples of Groups
- , the integers: .
- , the Rational numbers: .
- , the Real numbers: .
- , the non-zero Rational numbers: .
- , the non-zero Real numbers: .
- , the positive Rational numbers: .
- , the positive Real numbers:
In case that notation is confusing, the first is the group of Integers with addition as it’s operation; the last is the positive Real numbers with multiplication as it’s operation.
Now go lift something heavy,