Alternating Groups: Even Permutations in Sn for n > 2

[Gear300]

Alternating groups apply to all even permutations in S_n 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?

 

[jeffreydk]

If you go by the fact that the order of any alternating group is $n!/2$ then you would have that the order of $A_2$ is $2!/2=1$ 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 $A_2$

 

[Gear300]

If you go by the fact that the order of any alternating group is $n!/2$ then you would have that the order of $A_2$ is $2!/2=1$ 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 $A_2$

I see...so it would simply imply the identity element...Thanks

 

[jeffreydk]

No problem