Results 1 to 2 of 2

Math Help - Coset help

  1. #1
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303

    Coset help

    Q: Suppose that a has order 15. Find all of the left cosets of
    < a^{5} > in < a >.

    Proof. First, lets list the elements of < a >
    < a >=\{e, a, a^{2}, a^{3} , a^{4} , a^{5} , a^{6} , a^{7} , a^{8} , a^{9} , a^{10} , a^{11} , a^{12} , a^{13} , a^{14}\}

    The cosets would be:
    < a^{5} >, a < a^{5} >, a^{2} < a^{5} >, a^{3} < a^{5} >, a^{4} < a^{5} >

    I don't understand where the choices e, a, a^{2}, a^{3}, a^{4} came from. I see that there will be 5 cosets, since |<a^{5}>|||<a>|=\frac{15}{3}=5, but I don't understand the choices of the cosets.

    As I understand it, there will be more cosets, but cosets parition the group into distinct sets. So, once you have found all the cosets whos union is the orginal group, you are done. Correct?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by Danneedshelp View Post
    Q: Suppose that a has order 15. Find all of the left cosets of
    < a^{5} > in < a >.

    Proof. First, lets list the elements of < a >
    < a >=\{e, a, a^{2}, a^{3} , a^{4} , a^{5} , a^{6} , a^{7} , a^{8} , a^{9} , a^{10} , a^{11} , a^{12} , a^{13} , a^{14}\}

    The cosets would be:
    < a^{5} >, a < a^{5} >, a^{2} < a^{5} >, a^{3} < a^{5} >, a^{4} < a^{5} >

    I don't understand where the choices e, a, a^{2}, a^{3}, a^{4} came from. I see that there will be 5 cosets, since |<a^{5}>|||<a>|=\frac{15}{3}=5, but I don't understand the choices of the cosets.

    As I understand it, there will be more cosets, but cosets parition the group into distinct sets. So, once you have found all the cosets whos union is the orginal group, you are done. Correct?


    "So, once you have found all the cosets whos union is the orginal group, you are done. Correct?" Well, yes...obviously!

    The question here is to understand that a subgroup of a group defines an equivalence relation on the group

    whose equivalence classes are precisely the left (or right) cosets of the group wrt that sbgp.

    Not only that: if H\leq G\,,\,x,y,\in G\,,\,then\,\,xH=yH\Longleftrightarrow y^{-1}x\in H , so you could check easily whether

    two of the sets you were given are or not different cosets.

    For example, a<a^5>=a^3<a^5>\Longleftrightarrow a^{-3}a\in<a^5> . But a^{-3}a=a^{-2}=a^{13}\notin <a^5>\Longrightarrow the cosets

    are different.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Coset Representatives
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 30th 2010, 06:28 PM
  2. Coset Representatives
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: August 31st 2010, 04:15 AM
  3. Coset Decoding
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: May 25th 2010, 05:01 PM
  4. question about Coset
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 7th 2010, 12:16 AM
  5. Coset problem
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: September 7th 2008, 01:39 PM

Search Tags


/mathhelpforum @mathhelpforum