Answer:
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,
[tex](\alpha \cdot \beta)^n=\alpha^n \cdot \beta^n[/tex]
Proof:
Expanding from left, we have
[tex](\alpha \cdot \beta)^n=\alpha \cdot \beta \cdot \alpha \cdot \beta \cdot \alpha \cdot \beta ....\alpha \cdot \beta \:n-times[/tex]
Since α and β commutes, we rearrange to get:
[tex](\alpha\cdot \beta)^n=\alpha\cdot \alpha \cdot \alpha...n-times \cdot \beta \cdot \beta \cdot \beta-----n\:times[/tex]
We simplify on the right to get:
[tex](\alpha \cdot \beta)^n=\alpha^n \cdot \beta^n[/tex]
[tex]Q\cdot E\cdot D[/tex]
