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

< Abstract Algebra‎ | Group Theory‎ | Homomorphism

Contents

TheoremEdit

ProofEdit

IdentityEdit

0.   homomorphism maps identity to identity
1.   0. and  

2. Choose  
3.  
2.
4.  
i is in K and eK is identity of K(usage3)

5.   2, 3, and 4.
6.   is identity of   definition of identity(usage 4)

InverseEdit

0. Choose  
1.  
0.
2.  
homomorphism maps inverse to inverse between G and K
3.  
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.

ClosureEdit

0. Choose  
1.  
0.
2.  
Closure in G
3.  
4.  
f is a homomorphism, 0.
5.  
3. and 4.

AssociativityEdit

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