Embed Symmetric Group in Alternating Group A(n+2)

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In summary, the conversation discusses how to embed the symmetric group on n letters into the alternating group of (n+2) letters. The speaker is trying to come up with an explicit map but is unsure if a map that is the identity on An and not-the-identity elsewhere will work. The other person suggests using a permutation f(p) on U, where U is a set of n+2 symbols and p is a permutation in Sn. If p is an odd permutation, f(p) is defined as p * t, where t is an extra transposition (n+1, n+2). If p is an even permutation, f(p) is defined as p. It is shown that f is a homomorphism with
  • #1
How would you embed the symmetric group on n letters in the alternating group of (n+2) letters? I'm actually trying to write down an explicit map but can't seem to come up with one. I know An will be a subgroup of A(n+2) but I have a feeling that a map that is the identity on An and not-the-identity elsewhere won't work. Any thoughts?
 
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  • #2
Imagine a set U of n+2 symbols. A permutation p in Sn acts on the first n of these. If p is an odd permutation, define a new permutation f(p) on U by f(p) = p * t, where t is the extra transposition (n+1, n+2). Clearly, f(p) is in An+2. If p is an even permutation, define f(p) = p. It's easy to show that f is a homomorphism with trivial kernel.
 
  • #3
Thanks!
 

What is the definition of "Embed Symmetric Group in Alternating Group A(n+2)"?

Embedding a symmetric group in alternating group A(n+2) means showing how the elements of the symmetric group can be written as products of elements in the alternating group A(n+2).

What is the significance of embedding a symmetric group in alternating group A(n+2)?

Embedding a symmetric group in alternating group A(n+2) is significant because it allows us to simplify computations and proofs involving symmetric groups by using the properties of alternating groups.

How is an embedding of a symmetric group in alternating group A(n+2) represented?

An embedding of a symmetric group in alternating group A(n+2) is represented by a homomorphism from the symmetric group to the alternating group, which maps each element of the symmetric group to its corresponding element in the alternating group.

What are the conditions for an embedding of a symmetric group in alternating group A(n+2)?

For an embedding of a symmetric group in alternating group A(n+2) to exist, the order of the symmetric group must be less than or equal to the order of the alternating group, and the alternating group must contain all the even permutations of the symmetric group.

What are some real-world applications of embedding a symmetric group in alternating group A(n+2)?

Embedding a symmetric group in alternating group A(n+2) has applications in fields such as cryptography, coding theory, and group theory. It also has practical applications in areas such as computer science and engineering, where the properties of alternating groups can be used to simplify complex calculations and algorithms.

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