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 $(ab)^{mn}= e$ .

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) $\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 $a^t=e$ iff n is a divisor of t)

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

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

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

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

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) $\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 $a^t=e$ iff n is a divisor of t)

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

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

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

The trick is that in an abelian group, $(ab)^m=a^mb^m$...as simple as that!
Tonio

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

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) $\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 $a^t=e$ iff n is a divisor of t)

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

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

I have done all sorts of random inserts of $a^m$, $a^m * a^-m$ , etc. Nothing seems to give me what I need though... erm, I was wondering, since $a^m=e$ does $a^{-m}=e$?
1) $a^m=e, b^n=e$, which follows $a^{mn}=e, b^{mn}=e$. Since G is abelian, $(ab)^{mn}=a^{mn}b^{mn}=e$.
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 $a^{lcm(m,n)}=e$ and $b^{lcm(m,n)}=e$. Since G is abelian, it follows that $(ab)^{lcm(m,n)}=e$. Thus o(ab) | lcm(m,n) and o(ab)|mn.

3) Check 2 and 3 in $\mathbb{Z}_4$.

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### O(a)=m n O(b)=n in group then O(ab)=?

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