710: n-Symmetric Group

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definition of n-symmetric group

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Topics

About: group

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of group.

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Target Context

• The reader will have a definition of n-symmetric group.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$n$: $\in \mathbb{N}\setminus \{0\}$
$S$: $=\{1,...,n\}$
$\ast S_n$: $=\{\text{ the permutations on }S\}$ with the maps composition as the group operator
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Conditions:
//

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2: Natural Language Description

For any natural number, $n\in \mathbb{N}\setminus \{0\}$, and the set, $S:=\{1,...,n\}$, the set of the permutations on $S$, $S_n$, with the maps composition as the group operator

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3: Note

$S_n$ is indeed a group: 1) for any elements, $p_1,p_2,p_3\in S_n$, $(p_1{p}_2)p_3=p_1(p_2{p}_3)$, because $p_j$ is a map and composition of maps is associative; 2) the identity map, $id$, is the identity element; 3) for each element, the inverse map is the inverse element.

Of course, one can think of the permutations group of any other set of n-order, which is not exactly $S_n$ but is obviously 'groups - homomorphisms' isomorphic to $S_n$. We have specified $S$ as $\{1,...,n\}$ in order to uniquely determine $S_n$: very prevalently, the 'groups - homomorphisms' isomorphic groups are said to be "same", but we do no adopt that stance.

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References

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