Alternating Groups: Even Permutations in Sn for n > 2

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In summary, Alternating groups apply to all even permutations in Sn for n > 2, with the exception of A_2 which only consists of the identity element. This is because for n = 2, there are only two elements, resulting in the only transposition permutation being an odd permutation and therefore not in A_2.
  • #1
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]
 
  • #3
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
 
  • #4
No problem
 

1. What is an alternating group?

An alternating group is a type of mathematical group that consists of even permutations, or arrangements of objects, in a specific set called Sn. In simpler terms, it is a collection of even-numbered arrangements of a set of objects.

2. How are alternating groups represented?

Alternating groups are typically represented using cycle notation, where the elements in the set are arranged in cycles. For example, the alternating group A4 would be represented as (123)(4), which is the set of even permutations of the numbers 1, 2, 3, and 4.

3. What is the order of an alternating group?

The order of an alternating group A_n is n!/2, where n is the number of elements in the set. This is because for every even permutation, there is a corresponding odd permutation, resulting in half the total number of permutations.

4. How are alternating groups related to symmetric groups?

Alternating groups are a subgroup of symmetric groups, meaning that all alternating groups are also symmetric groups, but the reverse is not true. In other words, all alternating groups are subsets of symmetric groups.

5. What is the significance of alternating groups in mathematics?

Alternating groups have various applications in mathematics, particularly in group theory and combinatorics. They have properties that make them useful in solving problems related to symmetry, patterns, and permutations in different mathematical structures.

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