permutations help?

Question

a =
(1 2 3 4 5 6 )
(3 1 4 5 6 2 )
b =
(1 2 3 4 5 6)
(5 2 4 3 1 6)

Calculate ba^2

what is the order of a?

what is a^100?

Answer 1

The idea of multiplying permutations is this. To find ba, for example, we first perform the mapping b and follow it by the mapping a. Now b1 = 5 and a5 = 6, so ba1 = 6. Continuing we find ba =
(1 2 3 4 5 6)
(6 1 5 4 3 2).
For ba^2 = (ba)a , we start with ba as above and follow it with a.

If you start finding powers of a, you will find a^6 is the identity (so the order of a is 6). To find a^100, notice that 100 = 6*16 + 4, so a^100 = ((a^6)^16)a^4 = a^4, since (a^6)^16 is the identity permutation.

Answer 2

a = (3 1 4 5 6 2)
a^2 = (4 3 5 6 2 1)
ba^2 = (3 4 1 6 2 5)

a is of order six (it is a derangment on six)

a^100 mod 6 = 4 = (6 5 2 1 3 4)

Damn, I love group theory...