Let α by a cycle of length 3 and β a cycle of length 9. What is the order of αβ. Is the permutation αβ
even or odd?
b) Show that for every positive integer n we have
(αβ)^n = α^n*β^n

a) The order of  αβ is 9 and αβ is an even permutation

b) See proof

Step-by-step explanation:

a) Given that:  α is a cycle of length 3 and β a cycle of length 9.

We want to find the order of  αβ .

The order of αβ is the LCM of the length of the disjoint cycles.

Therefore the order is the LCM of 3 and 9 which is 9.

Since α is a cycle of length 3 and β a cycle of length 9, and (3-1)+(9-1)=10, the permutation αβ  is even.

b) We want to show that, for every positive integer n,

Proof:

Expanding from left, we have

Since  α and β commutes, we rearrange  to get:

We simplify on the right to get:

see below

step-by-step explanation:

x+2> 6

subtract 2 from each side

x+2-2 > 6-2

x > 4

there is an open circle at 4 since is it is greater than and the line goes to the right