After proving that the identity element of a group is unique, we next prove that the inverse of each element is also unique. This is one of the most frequently used facts in group theory. The group definition says that for each $a\in G$, there exists an inverse element $a^{-1}\in G$, but the definition does not initially say that the inverse is the only one. The uniqueness theorem allows us to write $a^{-1}$ without ambiguity. In this lesson, students will learn the formal proof of uniqueness of inverses, how inverse notation changes in additive and multiplicative groups, and why both left and right inverse equations are essential.

An inverse element depends on the element being inverted. In $(\mathbb{Z},+)$, the inverse of $7$ is $-7$, and the inverse of $-3$ is $3$. In $(\mathbb{R}-\{0\},\cdot)$, the inverse of $5$ is $\dfrac{1}{5}$, and the inverse of $-\dfrac{2}{3}$ is $-\dfrac{3}{2}$. The identity element is common to all elements, but inverses vary from element to element. This distinction is a common source of confusion in early group theory.

An inverse candidate must work from both sides. In additive modular arithmetic, every element has a unique inverse, but in multiplication modulo $n$ some elements may fail to have an inverse on the whole residue set. Choose an element and compare every possible candidate. A candidate is accepted only when both products equal the identity. This makes the two-sided inverse condition visible and helps separate group examples from non-group examples.

The proof uses associativity in the middle step. This is why uniqueness of inverse is a group property and not merely a property of a set with an identity. The expression $b\circ(a\circ c)$ is changed to $(b\circ a)\circ c$, and then the two inverse assumptions are used. Students often try to cancel $a$ before cancellation laws have been proved. At this point, cancellation should not be used as a shortcut because cancellation itself is proved later from inverses.

Use the calculator below to test inverse candidates in modular addition or modular multiplication. Choose a modulus $n$, an element $a$, and a candidate $b$. The calculator checks whether $a\circ b=b\circ a=e$ for the selected operation. This helps distinguish one-sided computation from the two-sided inverse condition in a group.