Wikipedia has separate pages for symmetric group and permutation group, but I don't understand what the difference between them is. A symmetric group on a set is the set of all bijections from the set to itself with composition of functions as the group action. Permutation group on a set is the set of all permutations of elements on the set.
Aren't these two things the same thing?
On one of the discussion pages, someone suggested that permutation groups don't have to include all permutations: they just have to be collections of permutations on the set, closed under composition etc. But this seems weird. First, that's not how I was taught it, and second (thanks to Cayley's theorem) it looks like it's redundant: all groups are "permutation groups" on this reading.
Is there some subtle difference I'm missing?
Best Answer
First, on the permutation group page, there's this line "the term permutation group is usually restricted to mean a subgroup of the symmetric group."
Second, Cayley's theorem doesn't really make the terminology "permutation group" redundant. When you talk about a permutation group, I think you are implicitly giving an action of the group on a set of some objects. This is an extra data other than the group structure.