# Permutation groups and symmetric groups

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?

-

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.

-
Right. It is important to make a philosophical distinction between permutation groups and abstract groups before one proves that the two are the same thing. Similarly, it is important to make a philosophical distinction between manifolds which are subsets of R^n and abstract manifolds before one proves that all abstract manifolds can be embedded into R^n. – Qiaochu Yuan Aug 28 '10 at 17:48

"Permutation group" usually refers to a group that is acting (faithfully) on a set; this includes the symmetric groups (which are the groups of all permutations of the set), but also every subgroup of a symmetric group.

Although all groups can be realized as permutation groups (by acting on themselves), this kind of action does not usually help in studying the group; special kinds of actions (irreducible, faithful, transitive, doubly transitive, etc), on the other hand, can give you a lot of information about a group. For example, Jordan proved that the only finite sharply five transitive groups are $A_7$, $S_6$, $S_5$, and the Mathieu group $M_{12}$. (A "sharply five transitive group" is a group $G$ acting on a set $X$ with five or more elements, such that for every ten elements $a_1,\ldots,a_5,b_1,\ldots,b_5\in X$, with $a_i\neq a_j$ for $i\neq j$ and $b_i\neq b_j$ for $i\neq j$, there exists one and only one $g\in G$ such that $g\cdot a_i = b_i$). (In fact, Jordan showed that the only finite sharply $k$-trasitive gruops for $k\geq 4$ are $S_k$, $S_{k+1}$, $A_{k+2}$, $M_{11}$, and $M_{12}$; see http://en.wikipedia.org/wiki/Mathieu_group.)

You think of a permutation group as a group $G$, together with a faithful action $\sigma\colon G\times X\to X$ on a set $X$ (faithful here means that if $gx=x$ for all $x$, then $g=e$). Cayley's Theorem tells you that every group $G$ can be thought of as a permutation group, by taking $X$ to be the underlying set of $G$, and $\sigma$ to multiplication. But this gives you an embedding of $G$ into a very large symmetric group, because the set on which it is acting is large. You usually get more information if the set you are acting on is "small"-ish.

The reason for Cayley's Theorem is that, historically, people only considered permutation groups: collections of functions that acted on sets (the sets of roots of a polynomial, the points on the plane via symmetries, etc). Cayley was trying to abstract the notion of group; he then pointed out that his more abstract definition certainly included all the things that people were already considering, and that in fact it did not introduce any new ones in the sense that every abstract group could be considered as a permutation group. But, as he pointed out, it is sometimes more convenient or useful to consider the group abstractly, sometimes to consider it as a group of permutations. Having both viewpoints is better than having just one. I think Cayley's Theorem has more historical interest than practical interest these days, but your mileage may vary.

-

A permutation group is a subgroup of a symmetric group on some set S.