Question

Let G be a finite abelian group and a, b belongs G with order (a) =m, order (b) =n. Which of the following are necessarily true?

1. Order (AB) =mn
2. Order (AB) = lcm(m,n)
3. There is an element of G whose order is lcm(m,n)
4. Order (AB) = gcd(m,n)​

Answer 1

Answer:

The answer is

4. Order (ab) = gcd(m,n)

Let c = gcd(m,n)

Then c is the smallest natural number such that

(ab) ^mm = e

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