Calculate the order in cycle notation [closed]

Please help calculate the order of x and y. Let x and y denote permutations of $N(7)$ - Natural numbers mod 7 . Cycle notation:

$$x= (15)(27436) $$

$$y= (1372)(46)(5)$$

Thanks

Solution 1:

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, as yours are, is equal to the least common multiple $(\operatorname{lcm})$ of the the orders of the cycles that form it, i.e., the least common multiple of the lengths of the disjoint cycles.

E.g. the order of $(1 2 3 4 5 6 7)$ is $7$. The order of $(123)(4567) = \operatorname{lcm}\,(3, 4) = 12$.

The order of $(123)(456)(7) = \operatorname{lcm}\,(3, 3, 1) = 3$.

Now, can you apply that to your permutations to find their orders?

Solution 2:

we start from left cycle. in the left cycle we have $1\to2$ and in the right cycle we have $2\to3$, so we deduce that $1\to3$.

now in the left cycle we have $3\to1$ and in the right cycle we have $1 \to 2$, so we deduce that 3$\to2$.

Finally in the left cycle we have$ 2\to3$ and in the right cycle we have $3 \to 1$, so we deduce that $2\to1$.

So we have in product of this two cycles $1\to3$ and $3\to2$ and $2\to1 $that is the cycle $(132)$.

Solution 3:

Hint: Find the least common multiple of the orders of each disjoint cycle.