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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 \( ag, ah \).