# Using group action to prove a set is a subgroup

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• Mr Davis 97
In summary, the point of this exercise is to introduce the concept of group actions and prepare for the orbit formula, which is an important theorem in the study of finite groups. By using group actions, it becomes easier to prove that ##G_i## is a subgroup of ##G## and find its order. This concept is also important for understanding Lagrange's theorem for finite groups.
Mr Davis 97
Problem: Let ##G=S_n##, fix ##i \in \{1,2, \dots, n \}## and let ##G_i = \{ \sigma \in G ~|~ \sigma (i) = i \}##. Use group actions to prove that ##G_i## is a subgroup of G. Find ##|G_i|##.

So here is what I did. Let ##A = \{1,2, \dots, n \}##. I claim that ##G## acts on ##A## by the group action ##\sigma \cdot i = \sigma (i)##. Proof: Let ##I_p## is the identity permutation. Then ##I_p \cdot i = I_p (i) = i##. Also, if ##\sigma_1, \sigma_2 \in G##, then it can quickly be checked that ##\sigma_1 \cdot (\sigma_2 \cdot i) = (\sigma_1 \circ \sigma_2) \cdot i##. This shows that we have a group action, which means that ##G_i##, the stabilizer of ##i## in ##G##, is automatically a subgroup. Also, by simple counting, we see that ##|G_i| = (n-1)!##.

Here is my real question: what was the point of this exercise? What's the utility of using the language of group actions when it seems I could have easily accomplished the same thing without it?

Mr Davis 97 said:
Problem: Let ##G=S_n##, fix ##i \in \{1,2, \dots, n \}## and let ##G_i = \{ \sigma \in G ~|~ \sigma (i) = i \}##. Use group actions to prove that ##G_i## is a subgroup of G. Find ##|G_i|##.

So here is what I did. Let ##A = \{1,2, \dots, n \}##. I claim that ##G## acts on ##A## by the group action ##\sigma \cdot i = \sigma (i)##. Proof: Let ##I_p## is the identity permutation. Then ##I_p \cdot i = I_p (i) = i##. Also, if ##\sigma_1, \sigma_2 \in G##, then it can quickly be checked that ##\sigma_1 \cdot (\sigma_2 \cdot i) = (\sigma_1 \circ \sigma_2) \cdot i##. This shows that we have a group action, which means that ##G_i##, the stabilizer of ##i## in ##G##, is automatically a subgroup. Also, by simple counting, we see that ##|G_i| = (n-1)!##.

Here is my real question: what was the point of this exercise? What's the utility of using the language of group actions when it seems I could have easily accomplished the same thing without it?
True. I also would have considered the direct way easier to see. The point of this exercise is to prepare for the - now I don't know the exact English term - we call it orbit formula. It is of similar importance as Lagrange's theorem for finite groups. It says:

Let ##(G,\cdot )## be a group which operates on a set ##M##, say ##(g,m) \longmapsto g.m \in M\,.##
Then we have for every ##m \in M## a bijection $$G/G_m \longleftrightarrow G.m = \{\,g.m\, : \,g \in G\,\} \subseteq M$$ where ##G_m## is the stabilizer and ##G.m## the orbit of ##m## under the operation of ##G##. Note that ##|G/G_m|## isn't necessarily a group, as the stabilizer may not be a normal subgroup. However, one can still build the equivalences classes ##g \,\cdot \,G_m##, it's just not necessarily a group again.

This yields for finite sets and groups ##|G/G_m| = |G.m|## resp. ##|G| = |G_m|\,\cdot \,|G.m|\,.##

Mr Davis 97

## What is group action?

Group action is a mathematical concept that describes how a group (a set of objects with a defined operation) can act on another set of objects, preserving the group structure.

## How can group action be used to prove a set is a subgroup?

If we can show that the operation of the larger group respects the structure of the smaller set, then the smaller set is a subgroup of the larger group. This can be done by demonstrating that the smaller set is closed under the operation of the larger group and that the identity element and inverse element of the larger group are also present in the smaller set.

## What is the significance of proving a set is a subgroup?

Proving a set is a subgroup is important because it helps us understand the structure and properties of a larger group. It also allows us to apply theorems and techniques from the larger group to the smaller set.

## Can group action be used to prove a set is not a subgroup?

Yes, group action can also be used to show that a set is not a subgroup. If the operation of the larger group does not respect the structure of the smaller set, then the smaller set is not a subgroup of the larger group.

## Are there any limitations to using group action to prove a set is a subgroup?

While group action is a powerful tool for proving subgroups, it may not always be the most efficient method. In some cases, other techniques such as direct proof or coset analysis may be more effective.

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