Thread: 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 .

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) 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 iff n is a divisor of t)

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

From what I understand, and

I have done all sorts of random inserts of , , etc. Nothing seems to give me what I need though... erm, I was wondering, since does ?

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 .

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) 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 iff n is a divisor of t)

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

From what I understand, and

I have done all sorts of random inserts of , , etc. Nothing seems to give me what I need though... erm, I was wondering, since does ?

The trick is that in an abelian group, ...as simple as that!
Tonio

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 .

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) 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 iff n is a divisor of t)

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

From what I understand, and

I have done all sorts of random inserts of , , etc. Nothing seems to give me what I need though... erm, I was wondering, since does ?

1) , which follows . Since G is abelian, .
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 and . Since G is abelian, it follows that . Thus o(ab) | lcm(m,n) and o(ab)|mn.

3) Check 2 and 3 in .