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

In summary, the alternating group in Sn is a subgroup of the symmetric group Sn that contains all the even permutations. It is important in Sn because it is the unique subgroup of index 2 and has implications in group theory and permutation theory. It is related to the symmetric group as a subset and the only normal subgroup. The order of the alternating group in Sn is n!/2, and it is defined as a subgroup that contains all even permutations or as the set of permutations that can be expressed as a product of an even number of transpositions.
  • #1
ToffeeC
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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?
 
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  • #2
An is simple. Try using that.
 

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

What is the alternating group in Sn?

The alternating group in Sn is a subgroup of the symmetric group Sn that contains all the even permutations. It is denoted by An and has half the order of Sn.

Why is the alternating group important in Sn?

The alternating group is important in Sn because it is the unique subgroup of index 2, meaning it is the only subgroup of Sn with half the order. This property has important implications in group theory and permutation theory.

How is the alternating group related to the symmetric group?

The alternating group is a subgroup of the symmetric group, meaning it is a subset of Sn that contains a smaller number of elements. It is also the only normal subgroup of Sn, meaning it is closed under conjugation by elements of Sn.

What is the order of the alternating group in Sn?

The order of the alternating group in Sn is n!/2, where n is the number of elements in Sn. This is because An contains half the number of elements of Sn, as it only contains even permutations.

How is the alternating group defined?

The alternating group is defined as a subgroup of Sn that contains all the even permutations. It can also be defined as the set of all permutations that can be expressed as a product of an even number of transpositions.

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