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:

the slope of the line is 60

one point on the line is (40,10)

step-by-step explanation:

in the equation outside the parenthesis the equation always tells you the slope. in this case it is 60.

in the parenthesis is also says 40 which is x because it is near the x and 10 is y because it is near y.

median is the value separating the higher half of a data sample, a population or probablity distribution from the lower half.

step-by-step explanation:

say you have 3,8,7,3,1,9,8 order them from lowest to highest

if orderd it would be 1,3,3,7,8,8,9 now you take away from the lowest and the highest until you end up in the middle.

so you would remove the 1 and 9 leaving you with 5 digits remaining

when you've reached the middle that would be your answer so in this case the answer is 7

if there were two numbers left say 7 and 5 you'd and them together so 12 then divide by 2 witch would equal 6 so the answer would be 6

good luck on that project hoped i you at all