411 REFERENCE · Wikibooks
Abstract Algebra/Group Theory/Homomorphism/Image 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 K.
Proof
Identity
{|Style = "width:60%;boarder:0.5px"
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|0.
|| homomorphism maps identity to identity
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|1.
|| 0. and
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|||
|-
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2. Choose ||
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3.
|| 2.
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4.
|| i is in K and eK is identity of K(usage3)
|-
|||
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|5.
|| 2, 3, and 4.
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|6. is identity of
|| definition of identity(usage 4)
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Inverse
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|0. Choose ||
|-
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1.
|| 0.
|-
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2.
|| homomorphism maps inverse to inverse between G and K
|-
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3.
|| homomorphism maps inverse to inverse
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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
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|0. Choose ||
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1.
|| 0.
|-
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2.
|| Closure in G
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3.
||
|-
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4.
|| f is a homomorphism, 0.
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5.
|| 3. and 4.
|}
Associativity
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| 0. im f is a subset of K||
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| 1. is associative in K||
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| 2. is associative in im f || 1 and 2
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