SOLUTION: Express as a product of transposition. a. [1 2 3 4 5 6 7 8 2 1 4 5 3 7 8 6 ] b. [1 2 3 4 5 6 7 8 3 7 5 8 6 1 4 2 ] c. [1 2 3 4 5 6 7 8 2

You have the permutations written wrong. 
You write them in two lines, like this.

a.

First we write the permutation as a product of cycles, then as product of
transposition.

1 goes to 2, 2 goes back to 1, where we started, so that's the cycle [1,2]
The first number you haven't been to is 3.
3 goes to 4, 4 goes to 5, 5 goes back to 3, where we started, so that's cycle [3,4,5]
The first number you haven't been to is 6.
6 goes to 7, 7 goes to 8, 8 goes back to 6, where we started, so that's cycle [6,7,8].
Now you've been to all the numbers 1 through 8 inclusive.
So the product of cycles is [1,2][3,4,5][6,7,8].

Now to change the product of cycles to a product of transposition:
A transposition is a cycle of length 2.

[1,2] is already a transposition because it has length 2.
[3,4,5] becomes [3,5][3,4]
[6,7,8] becomes [6,8][6,7][6,8]

So the product of transposition is [1,2][3,5][3,4][6,8][6,7]6,8]



b.

First we write the permutation as a product of cycles, then as product of
transposition.

1 goes to 3, 3 goes to 5, 5 goes to 6, 6 goes back to 1, where we started, so that's the cycle [1,3,5,6]
The first number you haven't been to is 2.
2 goes to 7, 7 goes to 4, 4 goes back to 8, 8 goes back to 2, where we started, so that's cycle [2,7,4,8]
Now you've been to all the numbers 1 through 8 inclusive.
So the product of cycles is [1,3,5,6][2,7,4,8].

Now to change the product of cycles to a product of transposition:
A transposition is a cycle of length 2.

[1,3,5,6] = [1,6][1,5][1,3]
[2,7,4,8] = [2,8][2,4][2,7]
So the product of transpositions is [1,6][1,5][1,3][2,8][2,4][2,7]


You do the third one the same way.  But write it like this first
Sometimes a cycle has just one number, when a number goes to itself. [Sometimes
people don't even write the cycles that have just one number.  They're just
understood.]

c.

Edwin

When I posted the solution above I only wrote the permutations as a product
of cycles.  I did not write the cycles as the product of transpositions. I
have edited the problem to show each cycle as a product of transpositions.

Transpositions are cycles of length 2. The first number in each transposition
is the first number in the cycle. The second number in each transposition is
the other numbers in the cycle in reverse order.

[a,b,c,d,e,f] = [a,f][a,e][a,d][a,c][a,b] 

Edwin