Proof the order of disjoint cycles is the least common multi

Proof the order of disjoint cycles is the least common multiple of the order of the cycles. (Give me some examples and solve them step by step )

Solution

Since you have the products of disjoint cycles, what do you know about the order of a cycle? For a single cycle, its order is equal to its length.

The order of a product of disjoint cycles is equal to the least common multiple (lcm)(lcm) of the the orders of the cycles that form it, i.e., the least common multiple of the lengths of the cycles.

E.g. the order of (1234567)(1234567) is 77. The order of (123)(4567)=lcm(3,4)=12(123)(4567)=lcm(3,4)=12.

The order of (123)(456)(7)=lcm(3,3,1)=3(123)(456)(7)=lcm(3,3,1)=3.