## Review of Group Theorem: Group Actions (Pt. I Definitions and a Sharpening of Cayley’s Theorem)

**Point of post: **In this post we discuss the ideas of group actions and the counting arguments one can use them for.

*Motivation*

We now move into one of the most beautiful subjects of finite group theory, the theory of group actions. Group actions occur everywhere in mathematics, most of the time without us even taking notice. Intuitively a group *acts *on a set by moving it’s members around in a specific way. For example, the permutation group *acts *on the set of labeled vertices of a triangle by moving them into a different configuration.

We shall see that group actions enable us to tell a lot about the structure of the underlying groups. In particular, it will give us the *class equation.*

*Group Actions: The Definitions*

Let be a group and a set. A mapping is called a *-action on if*

* *

* *

*Remark: *It is common, and we will follows this practice, to forgo the formality and denote by .

If there is a specified -action on we are apt to call a -space. The first thing that should be clear is:

**Theorem: ***Let be a -space with -action . Denote then the function given by . Then, the mapping is a homomorphism. Moreover, any homomorphism induces a -action on given by .*

**Proof: **Although obvious it is a good idea to indeed verify that is indeed a permutation on . To see that is injective we merely note that if then and by this implies that or and by we are now able to conclude that . To see that is surjective we merely note that for any we have that and

where we’ve used the same logic as in proving injectivity. Thus, . But, then the fact that is a homomorphism becomes trivial since for any and we have that

and thus as required.

Conversely, let be given by . Then, since is a homomorphism we have that

so that obeys . Also, since is a homomorphism we must have that so that for every , and so is also satisfied. The conclusion follows.

*Remark: *It’s clear from the fact that for we have that and so in particular if then .

If is a -space with -action and the associated homomorphism (as described in the last theorem) is injective we call the action *faithful.* We define the *kernel *of the -action to be the kernel of the associated homomorphism. More explicitly if the kernel of is defined to be the set

It’s clear from our alternate formulation of the kernel of the action that it is a normal subgroup of .

For we define the *isotropy subgroup of *, denoted , to be the set

In other words is the set of all elements of which *fix *. We now prove that the isotropy subgroup is worthy of its name:

**Theorem: ***Let be a -space with -action . Then, for any we have that .*

**Proof: **Let we note then that

and thus . The conclusion follows.

*Remark: *Notice that we used the fact that .

Lastly, if is a -space with -action then for we define the *orbit *of , denoted , to be the set

**References:**

1. Lang, Serge. *Undergraduate Algebra*. 3rd. ed. Springer, 2010. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 200

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Is there a standard technical name for the following concept of defining subsets of a group by “partial specification” of their actions?

For simplicity, consider the action of a group on itself. A group element g is thus identified as a function consisting of ordered pairs of elements of the group. Let S be a set of ordered pairs of group elements that don’t specify a many-to-one relationship. S may not contain enough ordered pairs to completely specify a function from the group onto itself. For example if the elements of the group are {a,b,c,d} the set S could be the ordered pairs { (a,b),(c,a) } which would not specify what a particular function had for (b,?) and (d,?).

It is possible to define a product operation on such specifications by saying that the product specification R = (S)(T) is set of ordered pairs that we can form by composing the maps defined by S and T insofar as that is possible with the ordered pairs they contain. Since it’s possible to do some algebra with partial specifications, I suspect they have already been studied, but I don’t recognize a standard topic in elementary group theory that fits.

Comment by Stephen Tashiro | October 2, 2012 |