True or False? If a and b are elements of an abelian group, then order of (ab)= lcm

**weijing85** · October 16th 2012, 06:23 AM

Dear friends,

May i check if the answer should be false?

True or false? If a and b are elements of an abelian group, then order of (ab)= lcm (order of (a), order of (b))

Thanks,
weijing

**johnsomeone** · October 16th 2012, 08:12 AM

ord(ab) and lcm(ord(a), ord(b)) have a divisibility relationship:

$(ab)^{lcm(ord(a), ord(b))} \ \overset{(abelian)}{=} \ (a)^{lcm(ord(a), ord(b))}(b)^{lcm(ord(a), ord(b))}$

$= (a)^{ord(a)k_1}(b)^{ord(b) k_2} = ((a)^{ord(a)})^{k_1} \ ((b)^{ord(b)})^{k_2} = (1)^{k_1}(1)^{k_2} = 1.$

so that ord(ab) always divides lcm(ord(a), ord(b).

But are they always equal? (Hint: think about inverses.)

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