suppose G is agroup and ab are in G for that ordG12 and ord6

suppose G is agroup and a,b are in G for that ord(G)=12 and ord(6)=15 if a and b commute , then ord(ab)25 ?

can ord(ab)=15 ?

note please solve each part for full steps solution regarding for the rule of abstract algebra

Solution

For commuting elements with relatively prime orders, the order of their product is the product of their orders. However, if the orders of two elements are not relatively prime, then the order of their product is the least common multiple (lcm) of the orders of the two elements. Here, ord(a) = 12 and ord(b) = 15. Since the lcm of 12 and 15 is 60, therefore ord(ab) = 60.