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{align*} ? &= ? \\ ? &= ? \end{align*}]

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:

[$? \implies g = h, \quad ? \implies g = h$]

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 $$ga, ha$$. Being epimorphic, $$a$$ doesn't allow any morphism to "hide" before $$a$$, like $$ag, ah$$.