Invertible Elements of Monoid form Subgroup of Cancellable Elements

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Theorem

Let $\struct {S, \circ}$ be an monoid whose identity is $e_S$.

Let $C$ be the set of all cancellable elements of $S$.

Let $T$ be the set of all invertible elements of $S$.


Then $\struct {T, \circ}$ is a subgroup of $\struct {C, \circ}$.


Proof

From Cancellable Elements of Monoid form Submonoid, $\struct {C, \circ}$ is a submonoid of $\struct {S, \circ}$.

Let its identity be $e_C$ (which may or may not be the same as $e_S$).

Let $T$ be the set of all invertible elements of $S$.

From Invertible Element of Monoid is Cancellable, all the invertible elements of $S$ are also all cancellable, so $T \subseteq C$.


Let $x, y \in T$.

Clearly $x^{-1}, y^{-1} \in T$, as if $x, y$ are invertible, then so are their inverses.


Taking the group axioms in turn:


Group Axiom $\text G 0$: Closure

By Inverse of Product, $x^{-1} \circ y^{-1} \in T$.

Thus $\struct {T, \circ}$ is closed.

$\Box$


Group Axiom $\text G 1$: Associativity

This is inherited from $S$, by Subsemigroup Closure Test.

Thus $\struct {T, \circ}$ is associative.

$\Box$


Group Axiom $\text G 2$: Existence of Identity Element

All the elements of $\struct {C, \circ}$ are by definition cancellable

So, by Identity of Cancellable Monoid is Identity of Submonoid:

$e_C \in T$

Thus $\struct {T, \circ}$ has an identity element.

$\Box$


Group Axiom $\text G 3$: Existence of Inverse Element

By Inverse of Inverse in Monoid:

$\struct {x^{-1} \circ y^{-1} }^{-1} \in T$

Thus every element of $\struct {T, \circ}$ has an inverse.

$\Box$


All the group axioms are thus seen to be fulfilled, and so $\struct {T, \circ}$ is a group.

$\blacksquare$


Sources

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