Results 1 to 3 of 3

Thread: Abelian group

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    1

    Abelian group

    a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then $\displaystyle (ab)^{mn}= e$ .

    b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).

    c) Give an example of an Abelian group G and elements a and b in G such that o(ab) $\displaystyle \neq\$o(a)o(b). Compare part (b)

    I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then $\displaystyle a^t=e$ iff n is a divisor of t)

    But, I just don't know what to do with part a...

    From what I understand, $\displaystyle a^m=e$ and $\displaystyle b^n=e$

    I have done all sorts of random inserts of $\displaystyle a^m$, $\displaystyle a^m * a^-m$ , etc. Nothing seems to give me what I need though... erm, I was wondering, since $\displaystyle a^m=e$ does $\displaystyle a^{-m}=e$?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by Zorae View Post
    a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then $\displaystyle (ab)^{mn}= e$ .

    b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).

    c) Give an example of an Abelian group G and elements a and b in G such that o(ab) $\displaystyle \neq\$o(a)o(b). Compare part (b)

    I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then $\displaystyle a^t=e$ iff n is a divisor of t)

    But, I just don't know what to do with part a...

    From what I understand, $\displaystyle a^m=e$ and $\displaystyle b^n=e$

    I have done all sorts of random inserts of $\displaystyle a^m$, $\displaystyle a^m * a^-m$ , etc. Nothing seems to give me what I need though... erm, I was wondering, since $\displaystyle a^m=e$ does $\displaystyle a^{-m}=e$?

    The trick is that in an abelian group, $\displaystyle (ab)^m=a^mb^m$...as simple as that!
    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by Zorae View Post
    a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then $\displaystyle (ab)^{mn}= e$ .

    b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).

    c) Give an example of an Abelian group G and elements a and b in G such that o(ab) $\displaystyle \neq\$o(a)o(b). Compare part (b)

    I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then $\displaystyle a^t=e$ iff n is a divisor of t)

    But, I just don't know what to do with part a...

    From what I understand, $\displaystyle a^m=e$ and $\displaystyle b^n=e$

    I have done all sorts of random inserts of $\displaystyle a^m$, $\displaystyle a^m * a^-m$ , etc. Nothing seems to give me what I need though... erm, I was wondering, since $\displaystyle a^m=e$ does $\displaystyle a^{-m}=e$?
    1) $\displaystyle a^m=e, b^n=e $, which follows $\displaystyle a^{mn}=e, b^{mn}=e$. Since G is abelian, $\displaystyle (ab)^{mn}=a^{mn}b^{mn}=e $.
    2) Let o(a)=m, o(b)=n. By using the basic number theory property, we see that mn=gcd(m,n)lcm(m,n). We know that $\displaystyle a^{lcm(m,n)}=e$ and $\displaystyle b^{lcm(m,n)}=e$. Since G is abelian, it follows that $\displaystyle (ab)^{lcm(m,n)}=e$. Thus o(ab) | lcm(m,n) and o(ab)|mn.

    3) Check 2 and 3 in $\displaystyle \mathbb{Z}_4$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Group, abelian
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 14th 2010, 05:05 AM
  2. abelian group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 16th 2010, 04:41 PM
  3. Is the subgroup of an abelian group always abelian?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Dec 6th 2009, 11:38 PM
  4. Abelian Group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 29th 2009, 05:38 AM
  5. Non- Abelian Group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Aug 25th 2008, 02:00 PM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum