Abstract Algebra/Group Theory/Homomorphism/Kernel 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 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 e_{G} 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 e_{G} 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