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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!) an 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 \( ag, ah \).