Group

Definition 1. A group is an invertible monoid, i.e. a magma $G$ such that:

(Associativity) For all $a$, $b$, $c \in G$,
\[(ab)c = a(bc).\]
(Unitality) There is one $e \in G$ such that for all $a \in G$,
\[ae = ea = a.\]
($e$ is automatically unique by NEU > Proposition 1.)
(Invertibility) For all $a \in G$, there is $x \in G$ such that
\[ax = xa = e.\]

Proposition 1. A monoid becomes a group as soon as it is left-invertible or right-invertible.

Proof. By MOID > Proposition 1.

Proposition 2. For any monoid $M$, its invertible subset $\inv M$ is a group.

Proposition 3. A group is uniquely invertible.

Proof. Let $G$ be a group with neutral element $e$ and let $a \in G$. If both $x$ and $x’$ are inverses of $a$,

\[x = xe = x(ax') = (xa)x' = ex' = x'.\]

Definition 2. In an abelian group $(G,+)$, we typically write $-a$ for the inverse of $a$ and define the subtraction

\[a - b = a + (-b).\]

Proposition 4. A group is cancellative.

Proof. Let $G$ be a group with neutral element $e$. Let $a$, $x$, $y \in G$.

If $ax = ay$, then
\[x = ex = (a^{-1}a)x = a^{-1}(ax) = a^{-1}(ay) = (a^{-1}a)y = ey = y.\]
If $xa = ya$, then
\[x = xe = x(aa^{-1}) = (xa)a^{-1} = (ya)a^{-1} = y(aa^{-1}) = ye = y.\]

\[(a^{-1})^{-1} = a.\]

\[(ab)^{-1} = b^{-1}a^{-1}.\]