Abstract Algebra/Group Theory/Homomorphism/Image 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 K.

Proof[edit]

Identity[edit]

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)

Inverse[edit]

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.

Closure[edit]

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

Associativity[edit]

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