I am studying Sylow theorems at the moment, more specifically trying to solve the following problem that I recently posted:
Let G be a finite group which has exactly eight Sylow 7 subgroups. Show that there exits a normal subgroup N of G such that the index [G:N] is divisible by 56 but not by 49
The link is here: If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.
My initial response was to use Sylow's theorems to understand the order of the group. Though I am still very new at this. I was then told to look at this problem through group actions, which I know considerably less in. After taking a look at group actions again, I know just a few more things. Considering that I am just beginning at understanding this theory, I was wondering if someone can explain to me in laymen terms, the application of group actions and how to use them to solve problems. Of course, I know that I could just look up book definition, but I would like a little more insight than that.
How would you explain group actions to someone with just calculus level knowledge maybe? What is the meaning of group actions, using language that is easy for a beginner to understand?
The definition of group action that I have found comes from Hungerford's Algebra. An action of a group $G$ on a set $S$ is a function $GxS \to S$ such that for all $x \in S$ and $g_1, g_2 \in G$: $ex=x$ and $(g_1g_2)x=g_1(g_2x)$
Best Answer
Note that another way to understand group actions is by "currying" the action into a function $G\to(S\to S)$; in this language a group action is just a homomorphism from $G$ to the permutation group on $S$. Thus in many ways the theory of group actions does not add anything more than you will get from understanding permutation groups and homomorphisms. Nonetheless, it can sometimes be useful to view group actions the "usual" way as a function $G\times S\to S$.
One great way to motivate the usage of group actions is to read the proof of Sylow's theorems. The "fundamental theorem" of group actions is the orbit-stabilizer theorem, and it is this that leads to most of the divisibility constraints in Sylow's theorems.