Difference between permutations

Question

Given the following:

1) Is it wrong to say (1 2 4) (5 3) = (1 2 4) (5 3) or = (3 5) (1 2 3) ?

2) What is meant by ( 1 2 3 4 5 ) and 1 2 3 4 5 ? And why are they not equal?

Thanks!

Answer 1

It seems that the notation $a_1a_2\ldots a_n$ is used for the permutation $i\mapsto a_i$ (I don't know the notation in this form, insstead I might have written such as $\begin{pmatrix}1&2&\cdots&n\\a_1&a_2&\cdots &a_n\end{pmatrix}$). Now to the problems: A cycle $(a_1\,a_2\,\ldots\, a_k)$ stands for the permutation that maps $a_1\mapsto a_2, a_2\mapsto a_3,\ldots,a_n\mapsto a_1$. Now note that disjoint cycles (i.e. having no element in common) commute.

For example in $(1\,2\,4)(3\,5)$, the right cycle maps $3\to 5$, the left cycle then leaves $5\mapsto 5$ unchanged. The same happens with $(3\,5)(1\,2\,4)$, where first $3$ is unchanged and then mapped to $5$. Therefore $(1\,2\,4)(3\,5)=(3\,5)(1\,2\,4)$.

Of course $(1\,2\,4)(3\,5)=(1\,2\,4)(3\,5)$ is trivially correct.

Now why is $(1\,2\,4)(5\,3)\ne(1\,4\,2)(5\,3)$? Note that the first maps $1\mapsto 2$ and the latter maps $1\mapsto 4$, so they are different.

By my introducing remark, I understand that $1\,2\,3\,4\,5$ stands for the permutation that maps $1\mapsto 1$, $2\mapsto 2$ and so on. But $(1\,2\,3\,4\,5)$ maps $1\mapsto 2$, $2\mapsto 3$ and so on, which is different.

$(1\,2\,3\,4\,5)$ and $2\,3\,4\,5\,1$ both denote the permutation that maps $1\mapsto 2$, $2\mapsto 3$, $3\mapsto 4$, $4\mapsto 5$, and $5\mapsto 1$.

Note that $(2\,3\,4\,5\,1)$ denotes the same permutation again, as it maps $2\mapsto 3$, $3\mapsto 4$, $4\mapsto 5$, $5\mapsto 1$, and $1\mapsto 2$, i.e. in effect the same map.

Answer 2

I can only answer your first: It's fine to say $(124)(35)=(35)(124)=(53)(124)=(53)(241)$, since $(124)$ and $(35)$ commute, i.e. they don't share any numbers, which means that they operate on different subsets of elements. Further you may cycle like $(124)=(241)$, but not change the overall order like $(124)\neq (142)$.