Michael Redman

Updated 2018 November 17

A group is a set S with an operation + and an identity element 0 such that:

- S is closed under +, meaning for any x and (not necessarily distinct) y in S, x+y is in S.
- + is associative, meaning for any a, b, c, in S, a+(b+c)=(a+b)+c
- For any x in S, x+0=x=0+x
- For any x in S, there exists in S an inverse of x, -x,such that x + (-x) = 0 = (-x) + x

By additive inverse property, there exists a -x with (-x)+x=0. Likewise by the additive inverse property there exists a -(-x) with -(-x)+(-x)=0. Then adding -(-x) to both sides of the first equation, -(-x)+(-x)+x=-(-x), implies x=-(-x).