Results 1 to 3 of 3

Thread: Prove quotient group cyclic

  1. #1
    Junior Member
    Joined
    Oct 2010
    From
    Zulu-5
    Posts
    60

    Prove quotient group cyclic

    Let $\displaystyle G=C_8\times C_8$ and $\displaystyle H\leq G$. $\displaystyle H$ is cyclic of order 8.

    Show that $\displaystyle G/H$ is cyclic.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member ModusPonens's Avatar
    Joined
    Aug 2010
    Posts
    125
    Thanks
    14

    Re: Prove quotient group cyclic

    I'm not sure my solution is correct, but here it goes.

    I assume $\displaystyle C_8$ is a cyclic group of order 8. So H is isomorphic $\displaystyle Z_8$, or to $\displaystyle \{0\} \times Z_8$. So $\displaystyle (C_8 \times C_8)/H$ is isomorphic to $\displaystyle (Z_8 \times Z_8)/(\{0\} \times Z_8)$, which is isomorphic to $\displaystyle (Z_8/ \{0\}) \times (Z_8/Z_8)$, which is isomorphic to $\displaystyle Z_8$, which is a cyclic group.

    Edited to, hopefuly, correct the mistakes.
    Last edited by ModusPonens; Sep 10th 2011 at 03:37 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2010
    From
    Zulu-5
    Posts
    60

    Re: Prove quotient group cyclic

    Thanks, You are correct. We have atheorem that states $\displaystyle (H\times K)/H \cong K$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove for cyclic group
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Jun 6th 2011, 10:47 AM
  2. Prove that group of order 15 is cyclic
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: Apr 10th 2011, 01:14 AM
  3. Prove that a quotient group is cyclic
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 8th 2010, 02:13 AM
  4. Prove cyclic subroups => cyclic group
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: Oct 11th 2009, 07:36 PM
  5. Prove a Group is NOT cyclic
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Feb 2nd 2008, 02:35 PM

Search Tags


/mathhelpforum @mathhelpforum