Composition Combination Of Permutations

Let be a set, and be a set of permutations of .

• If is a permutation of , we'll use exponential notation to express repeated composition of . This gives us and , etc.
• We'll use negative exponents when we want to repeated apply an inverse, which gives us .
• A composition combination of permutations in is a composition of the form where , each , and each for .
• The span of , written is the set of all composition combinations of permutations in . We'll say that the set generates .

As an example, if is a set of permutations of , then the composition is a composition combination of permutations in , and so are , , and more.

• Put them in a bulleted list.

definition.composition_combination_of_permutations.tex

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\begin{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. 
\begin{itemize}
\item We'll use negative exponents when we want to repeated apply an inverse, which gives us $\sigma^{-n}=\left(\sigma^{-1}\right)^n$.
\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}

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