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what does it mean by permutation commutes

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What does it mean by permutation commutes?

Permutation Groups

In abstract algebra, a permutation group is a group under the operation composition of function, generally represented by {eq}S_n, {/eq} when the set considered is {eq}S=\{1,2,3,\ldots,n\}. {/eq}

Answer and Explanation:

To be more specific and rigorous, let {eq}S {/eq} be a non-empty set and {eq}G = \{ f:S\to S: f \text{ bijetiva}\}. {/eq} If {eq}* {/eq} is the operation composition of functions {eq}* : G\times G \to G {/eq} where {eq}(g,f) \mapsto g\circ f. {/eq} {eq}(G,*) {/eq} is a group, and when we consider {eq}S=\{1,2,3,\ldots,n\}. {/eq} we can write {eq}S_n. {/eq}

The elements of {eq}S_n, {/eq} often does not commute with one another for {eq}n\geq 3. {/eq} We mean that a permutation commutes, when given two permutations {eq}\alpha, \beta \in S_n {/eq} we have that {eq}\alpha * \beta = \beta * \alpha. {/eq}


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