Q:

Composition on sets

[(1,2,3,4) ○ (1,2)(3,5)]^-1
The solution has the following steps:
f=(1,2,3,4) g=(1,2)(3,5)
Compute f○g:

Where can I find information regarding what the steps are doing in this problem? I understand how to compute the composition of two functions but I can't follow what's done for sets.

A:

Except for the odd style of writing things backward, this is the same as usual composition. $f \circ g$ means as usual first do $g$ then do $f$. For example $g=(12)(35)$ takes $1$ into $2$, then one sees what $f=(1234)$ takes that intermediate result $2$ into, and sees it is $3$. This shows $f \circ g$ carries $1$ to $3.$ If the cycle decomposition is desired the next one to do would be to see what $f \circ g$ takes $3$ into, in the above way, and so on to get the cycles for $f \circ g.$ After that is all done, the usual trick of reversing all the cycles in $f\circ g$ will give the cycle decomposition for $(f \circ g)^{-1},$ and if one likes one can shuffle the representations of the cycles now to place least elements first, etc.

