Abstract Algebra/Group Theory/Homomorphism/Kernel of a Homomorphism is a Subgroup
Appearance
Theorem
[edit | edit source]Let f be a homomorphism from group G to group K. Let eK be identity of K.
- is a subgroup of G.
Proof
[edit | edit source]Identity
[edit | edit source]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 | edit source]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 | edit source]0. Choose - 1.
0. - 2.
f is a homomorphism - 3.
1. and eK is identity of K - 4.
Associativity
[edit | edit source]0. ker f is a subset of G 1. is associative in G 2. is associative in ker f 1 and 2