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definition:permutation_group
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definition:permutation_group [2013/08/20 10:37] bmwoodruff |
definition:permutation_group [2013/08/30 11:47] (current) joshuawiscons [External links] |
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| ====== Permutation Group ====== | ====== Permutation Group ====== | ||
| ====Definition==== | ====Definition==== | ||
| - | Let $X$ be any set. A $\textdef{permutation group}$ on $X$ is any [[definition: | + | A $\textdef{permutation group}$ on $X$ is a set of [[definition: |
| - | + | ||
| - | ====Definition 2==== | + | |
| - | Let $X$ be any set. A permutation on $X$ is a bijection from $X$ to $X$. A $\textdef{permutation group}$ on $X$ is a set of permutations on $X$ that forms a group under function composition. | + | |
| ---- | ---- | ||
| ==== Remarks ==== | ==== Remarks ==== | ||
| - | * $\sym{X}$ denotes the [[definition: | + | * Alternately, |
| - | * The first definition requires that we first define symmetric group, subgroup, and that we have developed some intuition about how to show something is a subgroup. The second definition avoids this. | + | |
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| %%%%% | %%%%% | ||
| % DEPENDENCIES | % DEPENDENCIES | ||
| - | % RequiredMacros: | + | % RequiredMacros: |
| %%%%% | %%%%% | ||
| \begin{definition} | \begin{definition} | ||
| - | Let $X$ be any set. A \textdef{permutation group} on $X$ is any subgroup of $\sym{X}$; that is, it is a subset | + | A \textdef{permutation group} on $X$ is a set of permutations |
| \end{definition} | \end{definition} | ||
| </ | </ | ||
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| - | {{tag> | + | {{tag> |
definition/permutation_group.1377009452.txt.gz · Last modified: 2013/08/20 10:37 by bmwoodruff
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