Multiplication of transpositions? I can't seem to understand how the multiplication of two transpositions yield the results below:
$(x b)(x a) = (x a)(a b) \\
(c a)(x a) = (x c)(c a)$
I can't figure it out for the life of me. I'm trying to learn permutation multiplication on my own, and I feel confident in my abilities, but here I feel as though I'm missing something essential. Could someone please enlighten me as to the process (in depth) of multiplying these transpositions and getting the desired results?

Without loss of generality, suppose the only elements in the set are in fact $a,b,x$ for the first case.  Try writing each as a single permutation.
$(x~b)(x~a)$ read from right to left says:
$\begin{array}{c}a\mapsto x\mapsto b\\ x\mapsto a\mapsto a\\ b\mapsto b\mapsto x\end{array}$
I.e. $(x~b)(x~a) = \begin{pmatrix}a&b&x\\b&x&a\end{pmatrix}$
On the other hand, $(x~a)(a~b)$ read from right to left says:
$\begin{array}{c}a\mapsto b\mapsto b\\ b\mapsto a\mapsto x\\ x\mapsto x\mapsto a\end{array}$
So we have $(x~a)(a~b)=\begin{pmatrix} a&b&x\\b&x&a\end{pmatrix}$
These are indeed equal.  We have then $(x~b)(x~a)= (x~a)(a~b)$
Try reading through the second example to see if it is true.
