Which method is better for composing permutations in group theory?

[Gulli]

When composing permutations in group theory I don't know how to proceed. The way I see it there are two possible methods: assigning labels to the moving particles (e.g. "1" moves around the rhombus),or assigning labels to their starting positions (e.g. "1" is always the current particle at the top of the rhombus). The first methods would mean (1234)=(23)(21)(41) while the second methods would mean (1234)=(34)(24)(41). This seems very trivial but the two methods give the same result for many textbook examples so I've never been shown definitively which method to use.

 

[Nate810]

In group theory, the method of assigning labels to the moving particles is usually the way to go. This method is more commonly accepted because it allows for easier visualizations and makes it easier to keep track of the order of the permutations. It also makes it simpler to determine which permutation is being used for a particular problem. However, either method can be used depending on the individual's preference.