# Abstract Algebra

• May 6th 2006, 04:53 PM
mathlg
Abstract Algebra

Express (7 3 2 6) (5 7 1) (1 3 5 7) (4 6 8) as a product of transpositions
• May 6th 2006, 06:37 PM
ThePerfectHacker
Quote:

Originally Posted by mathlg

Express (7 3 2 6) (5 7 1) (1 3 5 7) (4 6 8) as a product of transpositions

The rule of transpositions is,
That a cycle,
$\displaystyle (a_n,a_{n-1},a_{n-2},...,a_2,a_1)$
Can be expressed as,
$\displaystyle (a_n,a_{n-1})(a_n,a_{n-2})...(a_n,a_1)$

I am confused by your question, are you asking to express that product as a transposition. Or is each cycle an individual problem. Your question is not well-defined.
• May 6th 2006, 07:22 PM
mathlg
Yes I am asking how to express that product as a transposition. Not each individual cycle.
• May 7th 2006, 08:27 AM
ThePerfectHacker
That big cycle which you mentioned is actually a premutation. I assume you are working in $\displaystyle S_8$
Let us express,
$\displaystyle (7,3,2,6) (5,7,1) (1,3,5,7) (4,6,8)$
As a premutation.
Since this is a function composition I will start with the inner function first, i.e. with (4,6,8). Then evaluate it by the second by the right and so one.

To find what premutation this is, be write out the mappings of each element from $\displaystyle S_8$.
Thus,
$\displaystyle \left\{ \begin{array}{c}1\to 1\to 3\to 3\to 2\\ 2\to 2\to 2 \to 2 \to 6\\ 3\to 3\to 5\to 7\to 3 \\ 4\to 6\to 6\to 6\to 7\\ 5\to 5\to 7\to 1\to 1\\ 6\to 8\to 8\to 8\to 8\\ 7\to 7\to 1\to 5\to 5\\ 8\to 4\to 4\to 4\to 4$
As a result we have the following premutation,
$\displaystyle \left( \begin{array}{cccccccc}1&2&3&4&5&6&7&8\\ 2&6&3&7&1&8&5&4 \end{array} \right)$
To express this premutation as a product of cycles, we need to find its orbits,
$\displaystyle 1\to 2\to 6\to 8\to 4\to 7\to 5\to 1$
Hence, the two equivalence classes are,
$\displaystyle \{1,2,6,8,4,7,5\} \mbox{ and } \{3\}$
Thus, this premutation is itself a cycle, since it has at most one orbit containing more than one element.
Thus, it can be expressed as,
$\displaystyle (1,2,6,8,4,7,5)$
Using the formula I stated above we can express it as a product of transpositions as,
$\displaystyle (1,2)(2,6)(6,8)(8,4)(4,7)(7,5)$