A group is an invertible monoid, i.e. a magma $G$ such that:
Associativity. For all $a$, $b$, $c \in G$,
\[(ab)c = a(bc).\]Neutral element. There is one $e \in G$ such that for all $a \in G$,
\[ae = ea = a.\]($e$ is automatically unique by NEU > Proposition 1.)
Inverse. For all $a \in G$, there is $x \in G$ such that
\[ax = xa = e.\]
A monoid becomes a group as soon as it is left-invertible or right-invertible.
Proof. By MOID > Proposition 1.
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 $a$,
\[x = xe = x(ax') = (xa)x' = ex' = x'.\]
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
\[x = xe = x(aa^{-1}) = (xa)a^{-1} = (ya)a^{-1} = y(aa^{-1}) = ye = y.\]
In an abelian group $(G,+)$, we write $-a$ for the inverse of $a$ and
\[a-b = a+(-b).\]
Proposition 4. Involutivity of Inversion.
Let $G$ be a group. For any $a \in G$,
\[(a^{-1})^{-1} = a.\]
Proposition 5. Antidistributivity of Inversion.
Let $G$ be a group. For any $a$, $b \in G$,
\[(ab)^{-1} = b^{-1}a^{-1}.\]
For any monoid $M$, $\inv M$ is a group.