Results 1 to 3 of 3

Thread: find the centre of S_n

  1. #1
    Senior Member abhishekkgp's Avatar
    Joined
    Jan 2011
    From
    India
    Posts
    495
    Thanks
    1

    find the centre of S_n

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

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

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

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

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

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

    If you have a different solution to the second one then please post it.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by abhishekkgp View Post
    question 1) Prove that $\displaystyle Z(S_n)=1 \text{ for all } n \geq 3$.

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

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

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

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

    This means either $\displaystyle |K: H \cap K|=1 \text{ or } p$
    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 $\displaystyle S_n\;\; n\geqslant 3$ 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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member abhishekkgp's Avatar
    Joined
    Jan 2011
    From
    India
    Posts
    495
    Thanks
    1
    Quote Originally Posted by Drexel28 View Post
    For the first one....don't think too hard. Given any element of $\displaystyle S_n\;\; n\geqslant 3$ 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???
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How do i find the centre of gravity
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: Jan 10th 2011, 10:59 AM
  2. Replies: 3
    Last Post: May 9th 2010, 03:22 PM
  3. Replies: 2
    Last Post: Apr 8th 2010, 07:43 PM
  4. Replies: 7
    Last Post: May 24th 2009, 07:41 AM
  5. Replies: 0
    Last Post: May 3rd 2009, 10:47 PM

/mathhelpforum @mathhelpforum