Answer to Question #196188 in Abstract Algebra for Mohd Arsalan

Question

Let τ be a fixed odd permutation in . S10 Show that every odd permutation in S10 is a product of τ and some permutation in A10

Answer 1

Solution:

We know that $A_{10}$ is the subset of even permutations in $S_{10}$.

$\tau$ is product of odd number of transpositions in $S_{10}$ ; permutation in $A_{10}$ is product of even number of transpositions in $S_{10}$ .

So, result of product of $\tau$ and permutation in $A_{10}$ is:

(odd number of transpositions in $S_{10}$ )+(even number of transpositions in $S_{10}$ )

=(odd number of transpositions in $S_{10}$ )

=odd permutation in $S_{10}$.