Abstract Algebra/Group Theory/Homomorphism/Image of a Homomorphism is a Subgroup

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|$\text{im}f=\{k\in K|\mathrm{\exists }g\in G:f(g)=k\}$]] is a [[../../Subgroup/Definition of a Subgroup|subgroup]] of K.

Proof

Identity

0. $f(e_G)=e_K$	[[../Homomorphism Maps Identity to Identity|homomorphism maps identity to identity]]
1. $e_K\in \{k\in K|\mathrm{\exists }g\in G:f(g)=k\}$	0. and [[../../Group/Definition of a Group/Definition of Identity|$e_G\in G$]]
2. Choose $i\in \{k\in K|\mathrm{\exists }g\in G:f(g)=k\}$||
3. $i\in K$	2.
4. $e_K\ast i=i\ast e_K=i$	[[../../Group/Definition of a Group/Definition of Identity#Usage3|i is in K and eK is identity of K(usage3)]]
5. $\mathrm{\forall }i\in \text{im}f:e_K\ast i=i\ast e_K=i$	2, 3, and 4.
6. $e_K$ is identity of $\text{im}f$	[[../../Group/Definition of a Group/Definition of Identity#Usage4|definition of identity(usage 4)]]

Inverse

0. Choose $i\in \{k\in K|\mathrm{\exists }g\in G:f(g)=k\}$
1. $\mathrm{\exists }g\in G:f(g)=i$	0.
2. $f(g)⊛f(g^{-1})=f(g^{-1})⊛f(g)=e_K$	[[../Homomorphism Maps Inverse to Inverse|homomorphism maps inverse to inverse]] between G and K
3. $i⊛f(g^{-1})=f(g^{-1})⊛i=e_K$	[[../Homomorphism Maps Inverse to Inverse|homomorphism maps inverse to inverse]]
4. i has inverse f( k-1) in im f	2, 3, and eK is identity of im f
5. Every element of im f has an inverse.

Closure

0. Choose $i_1,i_2\in \{k\in K|\mathrm{\exists }g\in G:f(g)=k\}$
1. $\mathrm{\exists }{g}_1,g_2\in G:f(g_1)=i_1,f(g_2)=i_2$	0.
2. $g_1\ast g_2\in G$	Closure in G
3. $f(g_1\ast g_2)\in \{k\in K|\mathrm{\exists }g\in G:f(g)=k\}$
4. $i_1⊛i_2=f(g_1)⊛f(g_2)=f(g_1\ast g_2)$	f is a homomorphism, 0.
5. $i_1⊛i_2\in imf$	3. and 4.

Associativity

0. im f is a subset of K
1. $⊛$ is associative in K
2. $⊛$ is associative in im f	1 and 2