Results 1 to 5 of 5

Math Help - homomorphisms and isomorphisms

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    3

    homomorphisms and isomorphisms

    I need help on this question!

    Suppose G is an Abelian Group and let f:G--> G be the function defined by f(x)=x^3

    i) show that f is a group homomorphism
    ii) show that f is an group homomorphism iff lGl is not divisible by 3

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by akvirdee View Post
    I need help on this question!

    Suppose G is an Abelian Group and let f:G--> G be the function defined by f(x)=x^3

    i) show that f is a group homomorphism
    ii) show that f is an group homomorphism iff lGl is not divisible by 3



    You must show (ab)^3=a^3b^3\,,\,\,\forall\,a,b\in G . in (ii) I think it must be "isomorphism", and then you must show a^3=1\Longleftrightarrow a=1 . Try it and write back if you get stuck somewhere.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2010
    Posts
    3
    thankyou for your help..the 1st part is pretty straight forward, however I did make a mistake with the second part it states

    ii) show that f is an isomorphism iff lGl is not divisible by 3.

    I don't quite understand how showing a^3=1 => a=1 shows that lGl is not divisible by 3.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by akvirdee View Post
    thankyou for your help..the 1st part is pretty straight forward, however I did make a mistake with the second part it states

    ii) show that f is an isomorphism iff lGl is not divisible by 3.

    I don't quite understand how showing a^3=1 => a=1 shows that lGl is not divisible by 3.
    If it is true that for all a\in G\,,\,\,a^3=1\Longrightarrow a=1 , then the group has no element of order 3 and thus its order cannot be divisible by 3 (read about Cauchy Theorem).

    Tonio
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Mar 2010
    Posts
    3
    Quote Originally Posted by tonio View Post
    If it is true that for all a\in G\,,\,\,a^3=1\Longrightarrow a=1 , then the group has no element of order 3 and thus its order cannot be divisible by 3 (read about Cauchy Theorem).

    Tonio



    Thank you for your help
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Homomorphisms/Isomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: October 22nd 2011, 07:40 PM
  2. Isomorphisms
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: July 11th 2011, 10:56 PM
  3. isomorphisms......
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 27th 2010, 10:11 PM
  4. Z2 + Z2 + Z3 isomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 2nd 2008, 06:34 PM
  5. isomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 30th 2008, 06:20 PM

Search Tags


/mathhelpforum @mathhelpforum