inverse

Inverse Element

$$\forall a\in A:\exists a^{\prime }:a\circ a^{\prime }=a^{\prime }\circ a=e$$

The inverse element is commutative and yields the neutral element $e$.
It is often denoted as $a^{-1}$ (for multiplicative groups) or $-a$ (for additive groups).

$(A,\circ )$ with associativity $⟹$ the inverse element is unique (every $x\in A$ has at most one inverse element $x^{\prime }$)

Proof: Let $a^{\prime },a^{\prime \prime }$ be inverses of $a$.

$$a^{\prime }=a^{\prime }\circ e=a^{\prime }\circ (a\circ a^{\prime \prime })=(a^{\prime }\circ a)\circ a^{\prime \prime }=e\circ a^{\prime \prime }=a^{\prime \prime }$$

Since we used associativity, this proof is only valid for associative structures.
→ There can be algebraic structures with multiple inverse elements, e.g. the quaternions.