Consider the following direct products Find the order of the

Consider the following direct products. Find the order of the given element in the group: G = Z_10 oplus U(13) Find the order of the element (4, 4) G= S_3 oplus Z_15 Find the order of the element ((123), 3 G= U(11) oplus U(25) Find the order of the element (4, 3)

Solution

a). Order of 4 in Z₁₀ is 5 and order of 4 in U(13) is 6. So the order of (4,4) in G will be least common multiple of 5 and 6 which is 30.

b). Order of (123) in S₃ is 3 and order of 3 in Z₁₅ is 5. So, the order of element ((123),3) in G is least common multiple of 3 and 5 which is 15.

c). Order of 4 in U(11) is 5 and order of 3 in U(25) is 20. So, the order of the element (4,3) in G is least common multiple of 5 and 20 which is 20.