Results 1 to 2 of 2

Thread: Finite Group,

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    3

    Finite Group,

    Let G be a finite group and let H be a non-empty subset of G that is closed under multiplication.Show that H is closed under inverse and hence prove that H is a subgroup of G.
    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 student17989 View Post
    Let G be a finite group and let H be a non-empty subset of G that is closed under multiplication.Show that H is closed under inverse and hence prove that H is a subgroup of G.
    Clearly since $\displaystyle |G|<\infty$ and $\displaystyle H\leqslant G$ we have that $\displaystyle |H|<\infty$. Let $\displaystyle h_0\in H$ be fixed and define $\displaystyle \theta_{h_0}:H\to H$ by $\displaystyle h\overset{\theta_{h_0}}{\longmapsto}h_0h$. Clearly this is injective since $\displaystyle \theta_{h_0}(h)=\theta_{h_0}(h')\implies h_0h=h_0h'$ and since $\displaystyle G$ is a group we have cancellation and thus $\displaystyle h=h'$. But, every injection from a finite set to itself is a bijection. Thus, since $\displaystyle e\in H$ we have that $\displaystyle \theta_{h_0}(h)=h_0h=e$ for some $\displaystyle h\in H$. It follows that $\displaystyle h=h_0^{-1}$ and since $\displaystyle h_0$ was arbitrary the conclusion follows.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finite group
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Oct 7th 2011, 02:48 PM
  2. [SOLVED] G is a finite group
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Sep 9th 2011, 07:52 PM
  3. Replies: 7
    Last Post: Feb 19th 2011, 03:29 PM
  4. finite group
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Dec 31st 2009, 10:49 AM
  5. Help with finite p-group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Dec 12th 2008, 06:42 AM

Search Tags


/mathhelpforum @mathhelpforum