(New page: <math>\scriptstyle\phi</math> is a homomorphism from <math>\scriptstyle G</math> to <math>\scriptstyle H</math>. Thus, we know that <math>\scriptstyle\phi(ab) = \phi(a)\phi(b) | a,b\in G</...) |
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<math>\scriptstyle\phi</math> is a homomorphism from <math>\scriptstyle G</math> to <math>\scriptstyle H</math>. Thus, we know that <math>\scriptstyle\phi(ab) = \phi(a)\phi(b) | a,b\in G</math>. <math>\scriptstyle\sigma</math> is a homomorphism from <math>\scriptstyle H</math> to <math>\scriptstyle K</math>. Thus, we know that <math>\scriptstyle\sigma(ab) = \sigma(a)\sigma(b) | a,b\in H</math>. | <math>\scriptstyle\phi</math> is a homomorphism from <math>\scriptstyle G</math> to <math>\scriptstyle H</math>. Thus, we know that <math>\scriptstyle\phi(ab) = \phi(a)\phi(b) | a,b\in G</math>. <math>\scriptstyle\sigma</math> is a homomorphism from <math>\scriptstyle H</math> to <math>\scriptstyle K</math>. Thus, we know that <math>\scriptstyle\sigma(ab) = \sigma(a)\sigma(b) | a,b\in H</math>. | ||
| − | Then, <math>\scriptstyle\sigma\phi(ab) = \sigma(\phi(ab)) = \sigma(\phi(a)\phi(b)) = \sigma(\phi(a))\sigma(\phi(b)) = \sigma\phi(a)\sigma\phi(b)</math>. We've shown that the group operation is preserved in a mapping <math>\scriptstyle\sigma\phi</math> from <math>\scriptstyle G</math> to <math>\scriptstyle K</math>, so <math>\scriptstyle\sigma\phi</math> is therefore a homomorphism from <math>\scriptstyle G</math> to <math>\scriptstyle K</math>. | + | Then, <math>\scriptstyle\sigma\phi(ab) = \sigma(\phi(ab)) = \sigma(\phi(a)\phi(b)) = \sigma(\phi(a))\sigma(\phi(b)) = \sigma\phi(a)\sigma\phi(b)</math>. We've shown that the group operation is preserved in a mapping <math>\scriptstyle\sigma\phi</math> from <math>\scriptstyle G</math> to <math>\scriptstyle K</math>, so <math>\scriptstyle\sigma\phi</math> is therefore a homomorphism from <math>\scriptstyle G</math> to <math>\scriptstyle K</math>. <math>\scriptstyle\Box</math> |
:--[[User:Narupley|Nick Rupley]] 21:02, 1 October 2008 (UTC) | :--[[User:Narupley|Nick Rupley]] 21:02, 1 October 2008 (UTC) | ||
Latest revision as of 17:06, 1 October 2008
$ \scriptstyle\phi $ is a homomorphism from $ \scriptstyle G $ to $ \scriptstyle H $. Thus, we know that $ \scriptstyle\phi(ab) = \phi(a)\phi(b) | a,b\in G $. $ \scriptstyle\sigma $ is a homomorphism from $ \scriptstyle H $ to $ \scriptstyle K $. Thus, we know that $ \scriptstyle\sigma(ab) = \sigma(a)\sigma(b) | a,b\in H $.
Then, $ \scriptstyle\sigma\phi(ab) = \sigma(\phi(ab)) = \sigma(\phi(a)\phi(b)) = \sigma(\phi(a))\sigma(\phi(b)) = \sigma\phi(a)\sigma\phi(b) $. We've shown that the group operation is preserved in a mapping $ \scriptstyle\sigma\phi $ from $ \scriptstyle G $ to $ \scriptstyle K $, so $ \scriptstyle\sigma\phi $ is therefore a homomorphism from $ \scriptstyle G $ to $ \scriptstyle K $. $ \scriptstyle\Box $
- --Nick Rupley 21:02, 1 October 2008 (UTC)
