Permutation groups and symmetric groups

Solution 1:

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.

Solution 2:

"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.