Alternating Groups: Even Permutations in Sn for n > 2

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Discussion Overview

The discussion centers on the properties of alternating groups, specifically A_n, and their relation to even permutations in the symmetric group S_n for n > 2. Participants explore the implications of these properties when n = 2, questioning the nature of permutations and the structure of the group.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant asserts that alternating groups apply to all even permutations in S_n for n > 2, raising a question about the case when n = 2.
  • Another participant calculates the order of A_2 as 1, concluding that it consists only of the identity element, thus implying that any transposition would be an odd permutation and not included in A_2.
  • A later reply reiterates the conclusion about A_2 being the trivial group, suggesting that this leads to the identity element being the only member.

Areas of Agreement / Disagreement

Participants generally agree that A_2 is the trivial group consisting only of the identity element. However, there is a debate regarding the implications of transpositions in this context and whether they can be classified as even or odd permutations.

Contextual Notes

The discussion does not resolve the implications of the properties of permutations in relation to the structure of alternating groups, particularly for small values of n.

Gear300
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Alternating groups apply to all even permutations in Sn for n > 2. Since n = 2 is inclusive, what got me wondering is that for such a case there are only 2 elements in S (say w and x); wouldn't that mean that the only transposition permutation would be (w x), which is an odd permutation?
 
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If you go by the fact that the order of any alternating group is [itex]n!/2[/itex] then you would have that the order of [itex]A_2[/itex] is [itex]2!/2=1[/itex] and therefore it's just the trivial group consisting of the identity element. Anything in the form of (wx) would be an odd permutation and therefore not in [itex]A_2[/itex]
 
jeffreydk said:
If you go by the fact that the order of any alternating group is [itex]n!/2[/itex] then you would have that the order of [itex]A_2[/itex] is [itex]2!/2=1[/itex] and therefore it's just the trivial group consisting of the identity element. Anything in the form of (wx) would be an odd permutation and therefore not in [itex]A_2[/itex]

I see...so it would simply imply the identity element...Thanks
 
No problem
 

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