# Abelian group

• Oct 20th 2009, 02:28 PM
Zorae
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\$?
• Oct 20th 2009, 10:15 PM
tonio
Quote:

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 \$\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
• Oct 20th 2009, 10:26 PM
aliceinwonderland
Quote:

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 \$\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\$.