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Math Help - find the centre of S_n

  1. #1
    Senior Member abhishekkgp's Avatar
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    find the centre of S_n

    question 1) Prove that Z(S_n)=1 \text{ for all } n \geq 3.

    question 2) Prove that if H \trianglelefteq G with  |G:H|=p,\text{ p is a prime}, then for all subgroups K of G either K \leq H or,
    G=HK \text{ and } |K:K \cap H|=p.

    I dont know how to solve the first one but i could solve the second one.
    here is my solution of the second one:

    since H \trianglelefteq G we find that HK = KH so HK \leq G.
    also H \trianglelefteq HK and HK/H \leq G/H so it makes sense to consider |G/H:HK/H|.

    Using |HK|= (|H||K|)/|H \cap K| we get,

    |G/H:HK/H| = |G:H||H|/|HK| = p|H \cap K|/|K| = p/|K: H \cap K|.

    This means either |K: H \cap K|=1 \text{ or } p
    this easily lads to the desired result.

    If you have a different solution to the second one then please post it.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by abhishekkgp View Post
    question 1) Prove that Z(S_n)=1 \text{ for all } n \geq 3.

    question 2) Prove that if H \trianglelefteq G with  |G:H|=p,\text{ p is a prime}, then for all subgroups K of G either K \leq H or,
    G=HK \text{ and } |K:K \cap H|=p.

    I dont know how to solve the first one but i could solve the second one.
    here is my solution of the second one:

    since H \trianglelefteq G we find that HK = KH so HK \leq G.
    also H \trianglelefteq HK and HK/H \leq G/H so it makes sense to consider |G/H:HK/H|.

    Using |HK|= (|H||K|)/|H \cap K| we get,

    |G/H:HK/H| = |G:H||H|/|HK| = p|H \cap K|/|K| = p/|K: H \cap K|.

    This means either |K: H \cap K|=1 \text{ or } p
    this easily lads to the desired result.

    If you have a different solution to the second one then please post it.
    For the first one....don't think too hard. Given any element of S_n\;\; n\geqslant 3 just construct something which doesn't commute with it. There are two more ways I can see to prove the second one, but the one you used is easiest.
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  3. #3
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Drexel28 View Post
    For the first one....don't think too hard. Given any element of S_n\;\; n\geqslant 3 just construct something which doesn't commute with it. There are two more ways I can see to prove the second one, but the one you used is easiest.
    i got the solution to the first one after i posted it.
    can you post your solutions of the second one???
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