Let G be Sn, the symmetric group of order n, acting as permutations on the set {1,2,...n}. Let H =?

Let G be Sn, the symmetric group of order n, acting as permutations on the set {1,2,...n}. Let H = {sigma element in G | n * sigma = n}.

(i) Prove that H is isomorphic to Sn-1.

(ii) Find a set of elements a1,...,an element in G such that Ha1,...,Han give all the right cosets of H in G.

(iii) Find the coset representation of G by H.

Not sure with proof. Please help!

2 Answers

  • ?
    Lv 5
    7 years ago
    Best Answer

    (i) This is trivial, but writing it down is fiddly. It would look better with bold chracters, but I'll make do with apostrophes. Please read X' as "The symbol X in boldface".

    I'll let Sn-1 act as permutations on the set { 1', 2', ..., (n-1)' } and for each s ∈ H define the function s' : { 1', 2', ..., (n-1)' } as follows:

    s'(k') = s(k).

    You can prove that s' is a bijection (permutation) as follows:

    s'(k') = s'(l') <=> s(k) = s(l) <=> k = l, so s' is injective.

    For every l' ∈ { 1', ..., (n-1)' }, we have l ∈ { 1, ..., n-1 }, and so there is some k ∈ { 1, ..., n-1 } with s(k) = l, and so s'(k') = l'. This means that s' is surjective.

    Now to show that the map s -> s' is an isomorphism: Let s,t ∈ H, then

    (s' o t')(k') = s'(t'(k')) = s(t(k)) = (s o t)(k) = s(k) t(k) = s'(k') t'(k').

    (ii) For k ∈ { 1, ..., n-1 } set ak = (k n), and let an be the neutral permutation in G.

    (iii) The cosets are given by H(ak) = { s ∈ G : s(n) = k }. Note that

    H = H(an) = { s ∈ G : s(n) = n } as you wanted.

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