(iii). Assume G is abelian. If ord(a)= 15 and ord(c)= 5, what can ord(ac) equal?

ne z solution Assume G is abelian and a, b EG Then for any " (ab)n = an b? Now we use this result. If order of (a) = 15 then (a)= = e where e il and order of (C)=5 then (eB=e identity element in G. Now (ac3 = 92 = e.co (ach=ca → {lace = fc82² co te where e is identity in a laco again {cacss?? = {c{2} = ('5 = e (ac) 15 = e where n=15 is smallest integer such that lacss=e. order of cac) = 15 which is required. Hence