I'm reading about symmetric groups and permutation cycles and I got confused between cycle composition and cycle product. I'm not quite sure what is the difference between the two. Lets say I have e.g. the following permutations:
$$\tau=\begin{pmatrix} 1\ 2\ 3\ 4\ 5\\ 1\ 5\ 3\ 2\ 4 \end{pmatrix}\;\;\;\;\;\;\;\sigma=\begin{pmatrix} 1\ 2\ 3\ 4\ 5\\ 2\ 4\ 3\ 5\ 1 \end{pmatrix}.$$
What is the difference between $\tau \;\circ \;\sigma$ and $\tau\sigma$? How do I compute them? Note that I'm a beginner in the subject ;)
Thank you for your help!
Edit:
Subquestion:
What does this also mean:
$$(123456)=(16)(15)(14)(13)(12)$$
I don't understand this decomposition.
There is no difference between $\tau\circ \sigma$ and $\tau\sigma$. About $(123456)=(16)(15)(14)(13)(12)$
Left hand side is a permutation that is a single cycle sending 1 to 2, 2 to 3, . . .5 to 6 and finally wrapping around 6 to 1.
The rhs is product ('composition') of 5 permutations of special kind: (14) means the permutation that swaps 1 and 4 and keeps others intact. Reading should be right to left (as function compositions are done that way). That is we process each number one by one by sending them through the 5 swaps.
Proceesing 1:
(12) sends it to 2, then that 2 is not disturbed by subsequent swaps to the left so 1 goes to 2 is final
Processing 2: (12) swap sends 2 to 1, then the immediate left swap (13) sends that 1 in turn to 3. But 3 is not affected by the other swaps further to the right. So 2 is evenutally to 3.
Processing 3: The rightmost swap(12) does not bother about 3 and simply passes it to the next swap (13) which sends it to 1 which in turn is sent to 4 by the next swap (14). Then the next two swaps (15) and (16) do not disturb 4. So 3 is finally sent to 4
etc
Hope this clarifies.