Results 1 to 2 of 2

Thread: Quotient groups

  1. #1
    ux0
    ux0 is offline
    Junior Member
    Joined
    Oct 2009
    Posts
    58

    Quotient groups

    Let G be a finite group with $\displaystyle K \lhd G$. If $\displaystyle (|K|,[G : K])=1$, prove that K is the unique subgroup of G having order $\displaystyle |K|$


    I was thinking about approaching this problem by..

    If $\displaystyle H \leq G$ and $\displaystyle |H|=|K|$, and I figure out what happens to elements of $\displaystyle H$ in $\displaystyle G/K$ then I might be able to see something happen?.... Or maybe I'm just approaching it all wrong
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by ux0 View Post
    Let G be a finite group with $\displaystyle K \lhd G$. If $\displaystyle (|K|,[G : K])=1$, prove that K is the unique subgroup of G having order $\displaystyle |K|$


    I was thinking about approaching this problem by..

    If $\displaystyle H \leq G$ and $\displaystyle |H|=|K|$, and I figure out what happens to elements of $\displaystyle H$ in $\displaystyle G/K$ then I might be able to see something happen?.... Or maybe I'm just approaching it all wrong
    suppose $\displaystyle H \leq G$ with $\displaystyle |H|=|K|.$ we have $\displaystyle HK \leq G,$ because $\displaystyle K \lhd G.$ therefore $\displaystyle |G|=n|HK|=n\frac{|K|^2}{|H \cap K|},$ for some integer $\displaystyle n \geq 1.$ so $\displaystyle |H \cap K|[G:K]=n|K|$ and hence $\displaystyle [G:K] \mid n,$ because

    $\displaystyle \gcd([G:K],|K|)=1.$ let $\displaystyle n=m[G:K],$ for some integer $\displaystyle m \geq 1.$ then $\displaystyle |K| \geq |H \cap K|=m|K| \geq |K|.$ thus $\displaystyle |H \cap K| = |K|$ and hence $\displaystyle H=K,$ because $\displaystyle |H|=|K|.$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Quotient Groups - Infinite Groups, finite orders
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Aug 11th 2010, 07:07 AM
  2. Quotient Groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 4th 2010, 03:30 PM
  3. Help with Quotient Groups
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Nov 24th 2009, 12:08 AM
  4. Quotient groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Sep 24th 2009, 11:58 AM
  5. quotient groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Sep 22nd 2009, 03:19 AM

Search Tags


/mathhelpforum @mathhelpforum