Inverses: Every Action Can Be Undone
The fourth axiom says: for every element $a$, there exists an element $a^{-1}$ (the inverse of $a$) such that:
$$a \cdot a^{-1} = a^{-1} \cdot a = e$$
Combining any element with its inverse gives back the identity — the “do nothing” element. Every action can be reversed.
Inverses in familiar groups
| Group | Operation | Element | Inverse | Check |
|---|---|---|---|---|
| $\mathbb{Z}$ (integers) | $+$ | $5$ | $-5$ | $5 + (-5) = 0$ |
| $\mathbb{Q}^*$ (non-zero rationals) | $\times$ | $3$ | $\frac{1}{3}$ | $3 \times \frac{1}{3} = 1$ |
| $S_3$ (permutations) | $\circ$ | $(123)$ | $(132)$ | $(123) \circ (132) = e$ |
| Mod 5 | $+$ | $3$ | $2$ | $3 + 2 \equiv 0 \pmod{5}$ |
| Mod 5 | $\times$ | $2$ | $3$ | $2 \times 3 \equiv 1 \pmod{5}$ |
Uniqueness of inverses
Each element has exactly one inverse. Suppose $a$ has two inverses $b$ and $c$. Then:
$$b = b \cdot e = b \cdot (a \cdot c) = (b \cdot a) \cdot c = e \cdot c = c$$
So $b = c$. (Notice how this proof uses associativity and the identity axiom — the axioms work together.)
Self-inverse elements
Some elements are their own inverse: $a \cdot a = e$. These are called involutions.
- In $\mathbb{Z}$ under addition: only $0$ ($0 + 0 = 0$)
- In permutations: any transposition like $(12)$ — swap twice and you’re back
- In the Klein 4-group: every non-identity element is self-inverse
Click an element to highlight it. Its inverse is the element it combines with to give the identity. Self-inverse elements are marked with a ★.
Why inverses matter
Inverses are what separate groups from lesser algebraic structures. A monoid has closure, associativity, and identity — but no inverses. The natural numbers ${0, 1, 2, \ldots}$ under addition are a monoid: you can add but you can’t subtract your way back to 0 from 5.
Groups guarantee reversibility. This is why groups model symmetry — every symmetry operation (rotation, reflection, permutation) can be undone.