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--- definition:permutation_group [2013/08/19 11:25]
joshuawiscons
+++ definition:permutation_group [2013/08/30 11:47] (current)
joshuawiscons [External links]
@@ Line 1: / Line 1: @@
 ====== Permutation Group ======
 ====Definition====
-Let $X$ be any set. A $\textdef{permutation group}$ on $X$ is any [[definition:subgroup]] of $\sym{X}$; that is, it is a subset of $\sym{X}$ that contains the identity and is closed under composition and taking inverses.
+A $\textdef{permutation group}$ on $X$ is a set of [[definition:permutation]]s of $X$ that contains the identity permutation and is closed under function composition and taking inverses.
 ----
 ==== Remarks ====
-  * $\sym{X}$ denotes the [[definition:symmetric group]] on $X$.
+  * Alternately, a permutation group is a subgroup of $\sym{X}$, where $\sym{X}$ denotes the [[definition:symmetric group]] on $X$.
@@ Line 14: / Line 13: @@
 %%%%%
 % DEPENDENCIES
-% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}}  \DeclareMathOperator{\sym}{Sym}
+% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}}
 %%%%%
 \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 of $\sym{X}$ that contains the identity and is closed under composition and taking inverses.
+A \textdef{permutation group} on $X$ is a set of permutations of $X$ that contains the identity permutation and is closed under function composition and taking inverses.
 \end{definition}
 </file>
@@ Line 26: / Line 25: @@
-{{tag>definition needsreview rjosh}}
+{{tag>definition needsreview rben ben rjosh}}

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