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Let f be a homomorphism from group G to group K.
Let eK be identity of K.
is a subgroup of K.
0.
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homomorphism maps identity to identity
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1.
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0. and
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- 2. Choose
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- 3.
![{\displaystyle i\in K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed832ac3945ec7b90ba50600a4e20a057d25598)
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2.
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- 4.
![{\displaystyle {\color {OliveGreen}e_{K}}\ast i=i\ast {\color {OliveGreen}e_{K}}=i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc91f816f07e866363a579f7feb897ad8672ef4)
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i is in K and eK is identity of K(usage3)
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5.
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2, 3, and 4.
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6. is identity of
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definition of identity(usage 4)
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0. Choose ![{\displaystyle {\color {OliveGreen}i}\in \lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2d6567bb48257c916ef14027cca8c12350b5a5) |
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- 1.
![{\displaystyle \exists \;{\color {OliveGreen}g}\in G:f({\color {OliveGreen}g})={\color {OliveGreen}i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98d6d35321a26a1a1d8fa568ff2a7384698335ab)
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0.
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- 2.
![{\displaystyle f({\color {OliveGreen}g})\circledast f({\color {BrickRed}g^{-1}})=f({\color {BrickRed}g^{-1}})\circledast f({\color {OliveGreen}g})=e_{K}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2076c97651f3d3f85bb66bbce05dcea4ae2aef18)
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homomorphism maps inverse to inverse between G and K
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- 3.
![{\displaystyle {\color {OliveGreen}i}\circledast f({\color {BrickRed}g^{-1}})=f({\color {BrickRed}g^{-1}})\circledast {\color {OliveGreen}i}=e_{K}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11b10fd447f9a67f5cbf7f2edd31ccb9a9e4a518)
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homomorphism maps inverse to inverse
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- 4. i has inverse f( k-1) in im f
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2, 3, and eK is identity of im f
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5. Every element of im f has an inverse.
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0. Choose ![{\displaystyle i_{1},i_{2}\in \lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1cf1db3d3bbef153ac312b54a988cc20facb21d) |
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- 1.
![{\displaystyle \exists \;g_{1},g_{2}\in G:f(g_{1})=i_{1},f(g_{2})=i_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce85d9af42b11ac55a004ca2ca353c5222272f5)
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0.
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- 2.
![{\displaystyle g_{1}\ast g_{2}\in G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7188b8e874443e6c642e11516bf49939a3d7cb6e)
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Closure in G
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- 3.
![{\displaystyle f(g_{1}\ast g_{2})\in \lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace }](https://wikimedia.org/api/rest_v1/media/math/render/svg/81ce75d368091096a9af168ba0a5b0ba2935d582)
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- 4.
![{\displaystyle i_{1}\circledast i_{2}=f(g_{1})\circledast f(g_{2})=f(g_{1}\ast g_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc467d707fb53701521f80c13985f1c2249a020b)
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f is a homomorphism, 0.
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- 5.
![{\displaystyle i_{1}\circledast i_{2}\in imf}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53225de5e5f352f2eafa8cd8cb69dd3cd6ff9020)
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3. and 4.
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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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