Definition. Let G be a group, and let X be a set. A left action of G on X is a function satisfying:
(a) for all and .
(b) for all .
It's customary to write " " or " " for ; the equations above then become
If G acts on X, X is a G-set.
Lemma. If X is a G-set, there is a homomorphism , where is the group of permutations of X. Conversely, any homomorphism gives rise to a G-set structure on X.
Proof. Let X be a G-set. Define by
Since and , and are inverses. Therefore, is a bijection, so .
To see that is a homomorphism, note that
Conversely, suppose is a homomorphism. Define an action of G on X by
This verifies property~1.
Since is a homomorphism, . Hence, . This verifies property~2. Hence, X is a G-set.
Example. Any group G acts on any set X by defining for all and all .
Example. Let act on by translation. That is, if and , define . Now
Therefore, this is an action of on .
Make G into a group via matrix multiplication. G acts on by multiplication: e.g.
Example. acts on . And in general, acts on .
Definition. Let X be a G-set, and let . The orbit of x under the action is
The set of orbits of X under G is denoted .
If for some , the action is transitive. (In this case, X is said to be a transitive G-set.)
Lemma. X is a transitive G-set if and only if for all , there exists such that .
Proof. ( ) Since X is transitive, for some . Let and let , where . Then , so .
( ) Suppose that for all , there exists such that . Take any . If , then for some , so . Thus, , so , and X is a transitive G-set.
Definition. Let X be a G-set, and let . The isotropy group of x is
If for all , the action is free. (In this case, X is called a free G-set.)
Example. Let , the dihedral group of order 6, act on the vertices of an equilateral triangle. If denotes reflection through the line through vertex 1, then the isotropy group of vertex 1 is (and similarly for the other two vertices).
Lemma. Let X be a G-set, and let . Then .
Proof. First, , so . Suppose . Then , so . Therefore, .
Finally, suppose . Then
Hence, . Therefore, .
Example. Consider the translation action of on defined above. An orbit has the form
Thus, the space of orbits is equivalent to , the circle. (Of course, the two are actually isomorphic as groups!)
The action is free, since the only which fixes is 0. On the other hand, the action is not transitive: There is no integer translation carrying 0 to , for instance.
Example. Consider the action of the group
on the plane .
The isotropy groups of points under this action are:
Obviously, the action is not free.
The orbits of points under the action are:
Since there is more than one orbit, the action is not transitive.
The set of orbits looks like the first quadrant .
Example. Consider the action of on . The action is transitive: If , , the transposition carries i to j. (If , there's nothing to do.)
The isotropy groups are
Proposition. Let X be a G-set. Then X is partitioned by the orbits.
Proof. If , then . Hence, X is certainly the union of the orbits. I need to show that two orbits are either disjoint or identical.
Suppose that . Find . Then and for some . Hence, , so and .
Now let . Then . This proves . Likewise, if , then , so . Therefore, . Hence, two orbits which intersect nontrivially must in fact coincide. This proves that the orbits partition X.
There is a group-theoretic relationship between isotropy groups of points in the same orbit.
Proposition. Let X be a G-set, let , and let . Then
Proof. Suppose , so . Then , so . Hence, . This shows that .
Conversely, suppose , where . Since ,
This shows that , so . Therefore, .
Corollary. Let X be a G-set. If x and y are in the same G-orbit, then and are conjugate subgroups. In particular, .
The following result not only serves as a "counting theorem" in the finite case, but also says that up to equivalence, every G-set "looks like" a union of sets of cosets of G.
Theorem. Let X be a G-set, and let .
(a) There is a bijection .
(b) If G is finite, : the order of the orbit equals the index of the isotropy group.
Proof. Define by
I must check that is well-defined.
Suppose then that . Then , so . Hence, . Now
Therefore, is indeed well-defined.
I must check that is well-defined.
Suppose that . Then , so . This means that , or . It follows that
Hence, is well-defined.
Since and are inverses, they are both bijections, and and are in 1-1 correspondence.
The second statement follows immediately from the first.
Definition. Let X be a G-set, and let . The fixed point set of X under S is
If , then denotes the elements of X fixed by g: i.e.
Example. Consider the action of on the set . Here is a picture of the isotropy groups and the fixed point sets:
A in a position means that the element of which labels the row is fixed by the element of which labels the column.
Example. Consider the group of symmetries of the hexagon below:
Since the hexagon is not regular, the symmetry group has only 4 elements: the identity map (id), reflection across the x-axis ( ), reflection across the y-axis ( ), and rotation by (r). The symmetry group acts on the set of vertices by permutations.
This action is not transitive (unlike the action in the preceding example). The orbits are and .
The following "counting theorem",has important combinatorial applications; it relates the number of orbits under a group action to the size of the fixed point sets.
Theorem. (Burnside) Let X be a finite G-set, where G is a finite group. Then
Proof. Observe that if and only if . It follows that
Now just compute:
Example. Each side of a square may be colored red, green, or blue. Colors may be repeated, and the square may be "turned over". How many distinct squares are there?
Note that since the square may be "turned over", the following colored squares are considered the same:
I'll temporarily nail the square to the wall so it can't move. There are then ways to paint the square. Two such paintings are "the same" if one can be carried into the other by a symmetry of the square --- i.e., by an element of .
Thus, X is the set of paintings of the fixed square. acts by symmetries. An equivalence class of paintings is simply an orbit under this action, and I want the number of orbits. According to Burnside, I simply count the number of fixed-square paintings (elements of X) fixed by each element of G.
For example, , reflection about the x-axis, fixes paintings where the top and bottom sides have the same color. Each side can be a different color, so there are paintings fixed by .
The total number of fixed points is 168. Since , there are distinct paintings.
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