Find two elements a b in some group G such that oa n ob m

Find two elements a, b in some group G such that o(a) = n, o(b) = m, the order of ab is finite, and o(ab) notequalto 1cm[n, m].

Solution

Since o(a) = n ==> aⁿ = e (where e is identity)

and o(b) = m , ===> b^m = e

therefore aⁿb^m = e*e = e

Order of ab is finite.

suppose o(ab) = k say. then (ab)^k = e

a^k b^k = e = aⁿb^m  

Now multiply both sides by the a^-n

So it becomes , a^-na^k b^k = e = a^-naⁿb^m  

So a^k-nb^k = b^m

Next multiply both sides by the b^-m

So it becomes a^k-nb^k b^-m= b^mb^-m

a^k-nb^k-m = e

Which can be written as a^k-n b^k-m = e*e = e

So a^{k - n} = b^{m - k}

So we can find two elements a and b such that a^{k - n} = b^{m - k} .