Abstract Algebra/Group Theory/Homomorphism/Image of a Homomorphism is a Subgroup
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< Abstract Algebra  Group Theory  Homomorphism
Theorem[edit]
Let f be a homomorphism from group G to group K. Let e_{K} 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 e_{K} 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 e_{K} 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