Axioms for Groups

The simplest and most memorable axioms for groups that I know are:

c c=c
abc a(bc) = (ab)c
ab(c ca = b ∧ c ac = b)

The first axiom merely states that the group is not empty. Next we have associativity and then solvability, left and right.

Group axioms often assume and describe an identity. Choose some element b. Solve for i in ib = b. i is a left identity for b.
For any other group element d solve for c in bc = d. Multiply ib = b on the right by c and we have ibc = bc thus id = d. The left identity for b is a left identity for all d. This is for an arbitrary d and thus i is a universal left identity. We have used associativity by omitting parentheses.

We find a universal right identity j by analogous logic. Now consider ij. i = ij = j and we see that there is just one universal identity in a group which is both a left identity and a right identity. We henceforth reserve “i” for that identity.