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

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Theorem[edit]

Let f be a homomorphism from group G to group K. Let eK be identity of K.


is a subgroup of G.

Proof[edit]

Identity[edit]

0. homomorphism maps identity to identity
1. 0. and
.
2. Choose   where  
3.
2.
4.
k is in G and eG is identity of G(usage3)
.
5. 2, 3, and 4.
6. is identity of definition of identity(usage 4)

Inverse[edit]

0. Choose
1.
0.
2.
definition of inverse in G (usage 3)
3.
homomorphism maps inverse to inverse
4. k has inverse k-1 in ker f
2, 3, and eG is identity of ker f
5. Every element of ker f has an inverse.

Closure[edit]

0. Choose
1.
0.
2.
f is a homomorphism
3.
1. and eK is identity of K
4.

Associativity[edit]

0. ker f is a subset of G
1. is associative in G
2. is associative in ker f 1 and 2