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--- definition:composition_combination_of_permutations [2013/11/21 11:10]
tarafife created
+++ definition:composition_combination_of_permutations [2013/11/21 11:14] (current)
tarafife [$\LaTeX$ version]
@@ Line 2: / Line 2: @@
 ====Definition====
 Let $X$ be a set, and $S$ be a set of permutations of $X$.
-*If $\sigma$ is a permutation of $X$, we'll use exponential notation to express repeated composition of $\sigma$. This gives us $\sigma^2=\sigma\circ \sigma$ and $\sigma^5=\sigma\circ\sigma\circ\sigma\circ\sigma\circ\sigma$, etc.
+  *If $\sigma$ is a permutation of $X$, we'll use exponential notation to express repeated composition of $\sigma$. This gives us $\sigma^2=\sigma\circ \sigma$ and $\sigma^5=\sigma\circ\sigma\circ\sigma\circ\sigma\circ\sigma$, etc.
-*We'll use negative exponents when we want to repeated apply an inverse, which gives us $\sigma^{-n}=\left(\sigma^{-1}\right)^n$.
+  *We'll use negative exponents when we want to repeated apply an inverse, which gives us $\sigma^{-n}=\left(\sigma^{-1}\right)^n$.
-* A composition combination of permutations in $S$ is a composition of the form $$\sigma_1^{n_1}\circ\sigma_2^{n_2}\circ\cdots\circ \sigma_k^{n_k},$$ where $k\in\mathbb{N}$, each $\sigma_i\in S$, and each $n_i\in \mathbb{Z}$ for $i\in \{1,2,3,\ldots,k\}$.
+  * A composition combination of permutations in $S$ is a composition of the form $$\sigma_1^{n_1}\circ\sigma_2^{n_2}\circ\cdots\circ \sigma_k^{n_k},$$ where $k\in\mathbb{N}$, each $\sigma_i\in S$, and each $n_i\in \mathbb{Z}$ for $i\in \{1,2,3,\ldots,k\}$.
-* The span of $S$, written $\text{span}(S)$ is the set of all composition combinations of permutations in $S$. We'll say that the set $S$ generates $\text{span}(S)$.
+  * The span of $S$, written $\text{span}(S)$ is the set of all composition combinations of permutations in $S$. We'll say that the set $S$ generates $\text{span}(S)$.
 As an example, if $S=\{a,b,c,d\}$ is a set of permutations of $X$, then the composition $a^2\circ b^{-1}\circ c^3$ is a composition combination of permutations in $S$, and so are $d^{-3}$, $b^5\circ a^{-2}$, and more.
@@ Line 24: / Line 24: @@
 Let $X$ be a set, and $S$ be a set of permutations of $X$.
 *If $\sigma$ is a permutation of $X$, we'll use exponential notation to express repeated composition of $\sigma$. This gives us $\sigma^2=\sigma\circ \sigma$ and $\sigma^5=\sigma\circ\sigma\circ\sigma\circ\sigma\circ\sigma$, etc.
-*We'll use negative exponents when we want to repeated apply an inverse, which gives us $\sigma^{-n}=\left(\sigma^{-1}\right)^n$.
+\begin{itemize}
-* A composition combination of permutations in $S$ is a composition of the form $$\sigma_1^{n_1}\circ\sigma_2^{n_2}\circ\cdots\circ \sigma_k^{n_k},$$ where $k\in\mathbb{N}$, each $\sigma_i\in S$, and each $n_i\in \mathbb{Z}$ for $i\in \{1,2,3,\ldots,k\}$.
+\item We'll use negative exponents when we want to repeated apply an inverse, which gives us $\sigma^{-n}=\left(\sigma^{-1}\right)^n$.
-* The span of $S$, written $\text{span}(S)$ is the set of all composition combinations of permutations in $S$. We'll say that the set $S$ generates $\text{span}(S)$.
+\item A composition combination of permutations in $S$ is a composition of the form $$\sigma_1^{n_1}\circ\sigma_2^{n_2}\circ\cdots\circ \sigma_k^{n_k},$$ where $k\in\mathbb{N}$, each $\sigma_i\in S$, and each $n_i\in \mathbb{Z}$ for $i\in \{1,2,3,\ldots,k\}$.
+\item The span of $S$, written $\text{span}(S)$ is the set of all composition combinations of permutations in $S$. We'll say that the set $S$ generates $\text{span}(S)$.
 As an example, if $S=\{a,b,c,d\}$ is a set of permutations of $X$, then the composition $a^2\circ b^{-1}\circ c^3$ is a composition combination of permutations in $S$, and so are $d^{-3}$, $b^5\circ a^{-2}$, and more.
+\end{itemize}
 \end{definition}
 </file>
@@ Line 36: / Line 38: @@
-{{tag>definition}}
+{{tag>definition ben}}

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