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Alternating group is the unique subgroup of index 2 in Sn?

  1. Feb 6, 2010 #1
    Where n >= 2.

    Is this true or false? I only got so far:

    If K is a subgroup of index 2, then it's normal. K is normal in Sn, so it's a union of conjugacy classes. Also, since |An K| = |An| |K| / |An intersection K| = 1/2n! * 1/2n! / |An intersection K| <= n!, then 1/4n! <= |An intersection K|.

    I don't know how to come up with a counter-example or proof from there. Could anybody help?
  2. jcsd
  3. Feb 6, 2010 #2
    An is simple. Try using that.
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