Identity: The Element That Does Nothing
The third axiom says: there exists a special element $e$ (the identity) such that combining it with any element $a$ gives back $a$ unchanged:
$$e \cdot a = a \cdot e = a$$
The identity is the “do nothing” element.
Identity elements in familiar groups
| Group | Operation | Identity | Why |
|---|---|---|---|
| Integers $\mathbb{Z}$ | Addition | $0$ | $0 + a = a + 0 = a$ |
| Non-zero rationals $\mathbb{Q}^*$ | Multiplication | $1$ | $1 \times a = a \times 1 = a$ |
| Permutations $S_n$ | Composition | $e$ (do nothing) | Leaving everything in place, then applying $\sigma$, gives $\sigma$ |
| Invertible matrices | Multiplication | $I$ (identity matrix) | $IA = AI = A$ |
| Mod $n$ integers $\mathbb{Z}_n$ | Addition mod $n$ | $0$ | $0 + a \equiv a \pmod{n}$ |
Uniqueness of identity
Every group has exactly one identity element. Here’s why: suppose $e$ and $f$ are both identities. Then $e \cdot f = f$ (because $e$ is an identity) and $e \cdot f = e$ (because $f$ is an identity). So $e = f$.
This tiny proof is typical of group theory — short, elegant, and definitive.
Here’s the multiplication table for a group. The identity element is the one whose row and column perfectly reproduce the header. Can you spot it?
Why identity matters
The identity element gives every group a neutral starting point. Combined with inverses (the next axiom), it means every operation can be undone — the identity is what you get back when an element meets its inverse.