>Let $ \sigma_{2} = (4215)(3426)(5617)$. Show that permutation as products of, in pair, disjoint cycles and as product of transpositions. 
Let $ \sigma_{2} = (4215)(3426)(5617)$. Show that permutation as products of, in pair, disjoint cycles and as product of transpositions.

I am a beginner and I am not sure how to start. Any hint helps! Thanks.   
 A: First the decomposition as a product of disjoint cycles:  compute the final image of $1$, then the image of its image, and so on:


*

*$1\mapsto7$,

*$7\mapsto 5\mapsto 4$,

*$4\mapsto2\mapsto 1$.


So we have a first cycle: $\;(1\,7\,4)$.
Can you calculate now the cycle starting with $2\,$?
As to expressing a cycle as a product of transpositions, it is easy: consider, for instance, the cycle $(1\,4\,3\,2)$. Multiply  it on the left by the transposition $(1\,4)$:
$$(1\,4)(1\,4\,3\,2)=(4\,3\,2).$$
Multiply the result by $(4\,3)$:
$$(4\,3)(4\,3\,2)=(3\,2)$$
Thus we obtain
$$(4\,3)(1\,4)(1\,4\,3\,2)=(3\,2),$$
and using that a transposition is its own inverse:
$$(1\,4\,3\,2)=(1\,4)(4\,3)(3\,2),$$
from which you can infer an easy computation rule for the decomposition of a cycle as a product of transpositions.
A: Start with $1$ and see where it goes, by applying the permutations from right to left.  Get $1\to7\to4\to1$, so $(174)$.  Then $2\to6\to5\to3\to2$, so $(2653)$.  
Next one way of writing a cycle as a product of transpositions is, for instance:   $(1234567)=(17)(16)(15)(14)(13)(12)$.
