Difference between revisions of "Isomorphism"
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| − | Given two [[Group (mathematics)|group]]s <math>G,G'</math>, an '''isomorphism''' from <math>G</math> to <math>G'</math> is a [[function]] <math>\phi : G \to G'\,</math> such that <math>\phi</math> is a [[homomorphism]] and <math>\phi</math> is [[Bijection|bijective]]. | + | Given two [[Group (mathematics)|group]]s <math>G,G'</math>, an '''isomorphism''' from <math>G</math> to <math>G'</math> is a [[function]] <math>\phi : G \to G'\,</math> such that <math>\phi</math> is a [[homomorphism]] and <math>\phi</math> is [[Bijection|bijective]]. |
| − | Two groups <math>G,G'</math> are called '''isomorphic''' if an isomorphism from <math>G</math> to <math>G'</math> exists. | + | |
| + | |||
| + | Two groups <math>G,G'</math> are called '''isomorphic''' if an isomorphism from <math>G</math> to <math>G'</math> exists. | ||
<!--Apologies if the TeX and font size don't work well together. I have Firefox set to use a specified font to avoid a weird bug, so I'm not sure how it'll look normally. - CSGuy --> | <!--Apologies if the TeX and font size don't work well together. I have Firefox set to use a specified font to avoid a weird bug, so I'm not sure how it'll look normally. - CSGuy --> | ||
| − | ==See | + | ==See also== |
*[[Automorphism]] | *[[Automorphism]] | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Latest revision as of 14:29, July 13, 2016
Given two groups , an isomorphism from to is a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi : G \to G'\,} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is a homomorphism and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is bijective.
Two groups Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G,G'}
are called isomorphic if an isomorphism from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}
to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G'}
exists.