411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Homomorphism/Kernel of a Homomorphism is a Subgroup
Reference material by Wikibooks contributors. This retains its source identity and is not 411 news reporting.
Theorem
Let f be a homomorphism from group <G to group <K.
Let <eK be identity of <K.
is a subgroup of G.
Proof
Identity
{|Style = "width:75%;boarder:1px"
|-
|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
{|Style = "width:75%;boarder:1px"
|-
|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
{|Style = "width:75%;boarder:1px"
|-
|0. Choose ||
|-
|
1.
|| 0.
|-
|
2.
|| f is a homomorphism
|-
|
3.
|| 1. and eK is identity of K
|-
|
4.
||
|}
<Associativity
{| Style ="width:60%"
|-
| 0. ker f is a subset of G||
|-
| 1. is associative in G||
|-
| 2. is associative in ker f || 1 and 2
|}