I'd like to know how people prefer to write permutations, or elements of the symmetric group $S_n$ for $n\ge0$.
The most natural way to define a permutation in $S_n$ is as a bijection on the set $\{1,2,….,n\}$. Then the set of permutations (bijections) becomes a group under composition of maps. If $f,g\in S_n$ then there are two ways to define the composition $f\circ g$ depending on whether our functions act from the left or the right:
$$(x)(f\circ g) = ((x)f)g\quad\text{and}\quad (f\circ g)(x)=f(g(x)).$$
I think that the latter is by far the most common these days. Of course, in the first case I could just define $(f\circ g)(x)=g(f(x))$ but it really is a right action so it should be written this way.
In terms of multiplying permutations using cycle notation the two ways of writing composition correspond, respectively, to whether we read the cycles left-to-right or right-to-left. For example:
$$(1,2)(2,3) = \begin{cases}
(3,2,1), &\text{using the $(x)f$ convention},\\
(1,2,3), &\text{using the $f(x)$ convention}.
\end{cases}$$
To me it has always seemed more natural to read permutations from left-to-right, as in the first case, but this implicitly uses the less common convention for composition of maps.
So the question: do you prefer to read products of permutations, written as cycles, from left-to-right or right-to-left?
Best Answer
The GAP convention is to multiply permutations from the left to the right, i.e. $(1,2) \cdot (1,3) = (1,2,3)$, to write down each cycle with the smallest moved point first and to sort cycles in ascending order w.r.t. their smallest moved points. I think meanwhile this convention is a somewhat widely used standard.