morrowcosom said:
Homework Statement
1)Consider the permutation in S3 = ( 1 2 3 )
( 1 2 3 ) NOTE: the two pairs of parenthesis are
meant to be one pair that encases both rows
So
\begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3\end{pmatrix}
This is the
identity permutation. Normally it is written just as "i". If you really
want to write it as a product of transpositions, you could use (1 2)(1 2). That is, transpose 1 and 2, then transpose them again, going back to what you had originally. (1 3)(1 3) will also work as will (2 3)(2 3). Or could do
two "transpositions and back": (1 3)(1 3)(1 3)(2 3)(2 3). Since each "transposition and back" requires two transpositions, no matter how many times we do that, we will have an
even number of transpositions.
Write as a product of transpositions
Homework Equations
The Attempt at a Solution
(1 3) (1 2), which was wrong. I randomly plugged in numbers until I ended up with the correct solution of (1,3) (1,2) (1,3) which contradicts all the examples of transpositions I have seen, like (1 3 2 4) = (1 4) (1 2) (1 3).
How does one go about finding the solution to my problem?
No, (1 3)(1 2)(1 3) will not work. (1 3) takes (1 2 3) to (3 2 1), then (1 2) take it to (3 1 2), and finally, (1 3) to (1 3 2), not (1 2 3). Since the identity permutation is an
even permutation, it can only be written as a product of an
even number of transpositons, not 3.
The identity transformation is pretty trivial. Are you sure you have copied the problem correctly?
