951: n-Alternating Group

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definition of n-alternating 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: Note

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Starting Context

• The reader knows a definition of n-symmetric group.

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Target Context

• The reader will have a definition of n-alternating 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:
$n$: $\in \mathbb{N}\setminus \{0\}$
$S$: $=\{1,...,n\}$
$\ast A_n$: $=\{\text{ the even permutations on }S\}$ with the maps composition as the group operator, $\in \{\text{ the subgroups of }{S}_n\}$
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Conditions:
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2: Note

$A_n$ is indeed a subgroup of $S_n$: 0) for each elements, $a_1,a_2\in A_n$, $a_1{a}_2\in A_n$, because as $a_1$ and $a_2$ can be realized as some sequences of even transpositions, $a_1{a}_2$ can be realized as the concatenation of the sequences, which is a sequence of even transpositions; 1) the associativity holds, because it holds on the ambient $S_n$; 2) the identity map, $id$, is in $A_n$, because it can be realized as the sequence of 0 transposition; 3) for each element, the inverse element, which is the inverse map, is in $A_n$, because it can be realized by the reverse sequence of the transpositions, which is a sequence of even transpositions.

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References

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