Abstract Algebra/Group Theory/Homomorphism/A Homomorphism with Trivial Kernel is Injective

Theorem

Let f be a [[../Definition of Homomorphism, Kernel, and Image|homomorphism]] from [[../../Group/Definition of a Group|group]] G to [[../../Group/Definition of a Group|group]] K. Let e_K be [[../../Group/Definition of a Group/Definition of Identity|identity]] of K.

[[../Definition of Homomorphism, Kernel, and Image|$\text{ker}f=\{e_G\}$]] means f is injective.

Proof

0. Choose $x,y\in G$ such that $f(x)=f(y)$
1. $f(y\ast x^{-1})=f(y)⊛f(x^{-1})$	f is a homomorphism
2. $=f(x)⊛f(x^{-1})$	0.
3. $=f(x\ast x^{-1})$	f is a homomorphism
4. $=f(e_G)=e_K$	homomorphism maps identity to identity
5. $y\ast x^{-1}\in \text{ker}f$	1,2,3,4.
6. $y\ast x^{-1}=e_G$	given $\text{ker}f=\{e_G\}$
7. $y=x$