∧a∧b∧c a(bc) = (ab)c
∧a∧b(∨c ca = b ∧ ∨c ac = b)
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.