Section 17.1 Pre-read
A central philosophy of group theory is that a group should measure symmetry. So generally the structure of a group is best studied by pairing a group with an object on which the group elements ``act'' by way of symmetries. This object does not have to be a geometric object. Usually it is a set, and group elements act by permuting the set.
Example 17.1.1. \(D_3\) acts on an equilateral triangle.
We created the group \(D_3\) as the group of symmetries of an equilateral triangle. But it is more common to work with elements of \(D_3\) as permutations of the vertices of that triangle. That is, if we number the vertices of the triangle, then we usually treat \(D_3\) as a subgroup of \(S_3\text{,}\) even if we don't explicitly say so. (In fact, as we saw, \(D_3\) is isomorphic to \(S_3\text{.}\))
Example 17.1.2. \(\GL_n(\R)\) and its subgroups act on \(\R^n\).
A matrix times a column vector results in another column vector, so we can think of any group of invertible matrices as a group of geometric transformations of \(\R^n\text{.}\) That is, we can think of invertible matrices as elements of \(S_{\R^n}\text{.}\)
So, just as we did when we explored Cayley's Theorem, we want to relate a given group \(G\) to some subgroup of \(S_X\) for some set \(X\text{,}\) However, in Cayley's Theorem we took a very particular approach for a very particular purpose; now we will be much more general.
Definition 17.1.3. Group action.
A homomorphism from a group \(G\) to the group \(S_X\) of permutations on some set \(X\text{.}\)
If we have such a homomorphism \(\funcdef{\varphi}{G}{S_X}\text{,}\) then for \(g\) in \(G\) the image \(\varphi(g)\) is an element of \(S_X\text{,}\) so it is a function itself; in particular, each
is a bijection. It is usually too clumsy to use notation like \(\varphi(g)(x)\) to mean the result of interpreting an element \(g\) in \(G\) as a permutation of the set \(X\) and then applying that permutation to the specific element \(x\) of \(X\text{,}\) so we usually just interpret \(g\) itself as a function on \(X\) and write
instead. (Another common notation is to use multiplicative notation in analogy with the matrices-acting-on-vectors example and writing \(g \cdot x\text{,}\) but we will avoid this notation to avoid confusion with the group operation in \(G\text{.}\))
Here are two important concepts related to group actions. Suppose group \(G\) acts on set \(X\text{,}\) and \(x\) is some specific element in \(X\text{.}\)
Definition 17.1.4. Stabilizer of set element \(x\).
Write \(G_x\) or \(\Stab_G(x)\) to mean the collection of those group elements \(g\) so that \(\grpact{g}{x} = x\text{.}\)
Definition 17.1.5. Orbit of set element \(x\).
Write \(\grpact{G}{x}\) or \(\orbit{x}\) to mean the collection of all possible results of applying the group action on \(x\text{.}\) That is, an element \(y\) in \(X\) is in the orbit \(\grpact{G}{x}\) precisely when there is at least one \(g\) in \(G\) so that \(y = \grpact{g}{x}\text{.}\)
Warning 17.1.6. Keep it straight!
A stabilizer is a collection of group elements from \(G\text{,}\) while an orbit is a collection of set elements from \(X\text{.}\)