Math and science::Algebra::Aluffi

Groups. 3 basic lemmas.

This card covers three very basic and fundamental properties of groups.

The identity element is unique

If $h \in G$ is an identity of $G$ , then $h = e_{G}$ .

Proof. Let $h$ and $e_{G}$ be identities of $G$ . Then we have:

[

? .

]

The inverse is unique

If $h_{1}, h_{2}$ are both inverses of $g$ in $G$ , then $h_{1} = h_{2}$ .

Proof. Let $h_{1}, h_{2}$ be inverses of $g$ . Then we have:

[

\begin{aligned} ? & = ? \\ ? & = ? \end{aligned}

]

By [what?], $(h_{1} g) h_{2} = h_{1} (g h_{2})$ , so we must have $h_{1} = h_{2}$ .

Cancellation

Let $G$ be a group, and let $a, g, h \in G$ . The following holds:

[

? ⟹ g = h, ? ⟹ g = h

]

Both cancellation statements follow easily by composing $a^{- 1}$ and applying associativity. To appeal to intuition, note that (I think!) a isomorphism must be both monomorphic and epimorphic (be careful to note that the inverse implication doesn't hold). Being monomorphic, $a$ doesn't allow any morphism to "hide" after $a$ , like $g a, h a$ . Being epimorphic, $a$ doesn't allow any morphism to "hide" before $a$ , like $a g, a h$ .

09.11.2020