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I've been studying group theory for two months, and we're now on "Actions on a group G". The definition we were given was this:

Definition (Group action): Let $G$ be a group. Let $X$ be a set. We say that $G$ acts on the set $X$ if exists a map \begin{equation} G \times X \to X\\ (g,x)\to gx, \end{equation} such that:

($i$) $ex=x$, $\forall x \in X$

($ii$) $g(hx)=(gh)x$, $\forall g,h\in G, \forall x \in X$.

I tried to understand intuitively everything about group theory, but I'm not capable of understand this. Why group actions are important? Is there an intuitive way to understand them?

Thank you.

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Some examples of group actions:

$S_n$, the symmetric group on $n$ letters, naturally acts on the set $[n] = \{1, \ldots, n\}$. For example, given $\sigma = (1 3 2) \in S_4$, we have

\begin{align*} \sigma(1) &= 3 \\ \sigma(2) &= 1 \\ \sigma(3) &= 2 \\ \sigma(4) &= 4. \end{align*}

This is probably one of the first group actions you've learned, quite likely before you had even heard the term group action.

Another example: The group dihedral group $D_{2n}$ acts on the regular $n$-gon. Here are all the ways $D_6$ acts on a regular triangle (shamelessly borrowed from this website): enter image description here

Thus, when people say that group theory is a way to study the symmetries of an object, they're really talking about finding a suitable group to act on the object.

Given a group $G$ and an action of $G$ on an object $X$, we can define the stabilizer of $x \in X$ as $\text{Stab}_G(x) = \{g \in G : gx = x\}$. These are all the group elements that fix a particular $x \in X$. In the example of $D_6$ above, each vertex of the triangle is fixed by exactly two elements of $D_6$, the identity of $D_6$, and one reflection.

In a similar vein, we can also define the orbit of an element $x \in X$ as $\text{Orb}_G(x) = \{gx : g \in G\}$. These are all the different things in $X$ that a particular $x$ could get mapped to, when we let each group element $g \in G$ get a turn.

But wait, there's more! You remember conjugation, right? Given a group $G$ and two elements $g, x \in G$ we can conjugate $x$ by $g$, to get the element $x^g = g^{-1}xg$. You can show that a group acts on itself, via conjugation. This conjugation action is extremely important!

Under the conjugation action of $G$ on itself, given a fixed $x \in G$, what is the orbit $\text{Orb}_G(x)$? Unpacking the definition, $\text{Orb}_G(x) =\{g^{-1}xg: g \in G\}$ is nothing more than the conjugacy class of $x$! What is the stabilizer? $\text{Stab}_G(x) = \{g \in G: g^{-1}xg = x\}$, where $g^{-1}xg = x$ if and only if $xg = gx$. Thus, the stabilizer of $x$ is nothing more than the centralizer of $x$; everything that commutes with $x$!

It's not hard to see that we can extend the conjugation action of $G$ on itself to let $G$ act on the set of all subgroups of $G$. In this case, the normal subgroups of $G$ are precisely those subgroups with a trivial orbit. More generally, the stabilizer of a subgroup $H$ is the normalizer of $H$.

So, not only are group actions the main practical reason anyone besides group theorists learn about groups (because a group that isn't acting on anything could reasonably be called boring), but they can also unify a lot of purely group-theoretic ideas and constructs.

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Sometimes we don't need external goals or applications to motivate us to study something; we might find that something interesting in and of itself. Many nonmathematicians would feel that way about most of pure mathematics, especially the stuff which doesn't have any known real-world uses, is this thing to mathematicians.

But it would be incorrect to think that's all that can be said about group actions. Indeed if one thinks group theory is inherently interesting then one should feel similarly about group actions, since actions are to some extent the point of groups. Groups are to group actions as potential energy is to kinetic energy. A group by itself is just a bunch of (potentially nameless) things that can be composed together and inverted with the same mechanics as a set of functions, and actions are what those things do to other things. Actions are how the abstract algebraic structure of a group encodes the notion of symmetry to begin with (although it is this perspective that made me feel that unfaithful representations were odd and counterintuitive - if they aren't faithful, in what sense are they "representative" according to the colloquial understanding of the word?).

Historically groups (before they had that name) were first conceived of as permutation groups, which are groups of functions, in the context of symetries of roots of polynomials. It is only later that the notion was formally abstracted into an algebraic structure that made no definitional references to functions. Cayley's theorem brings actions back into the picture in a sense because it says any group can be viewed as a permutation group, but different objects can have the same symmetry groups, so it makes sense to allow an abstract group to act on different things instead of necessarily defining different isomorphic concrete groups for wherever they might manifest. In this way we can compare the symmetry of different objects, and we can also speak of induced actions (when the action of a group on one object naturally leads to an action of it on some kind of object which is constructed/derived from or otherwise associated with the first).

There are many natural types of group actions, usually where the actions are symmetries which in some sense preserve the structure or features of the object being acted on. In Galois theory we study actions which preserve the truth of all (polynomial) equations, for which it is sufficient for a map to simply preserve addition and multiplication. Thus Galois groups are the symmetries of number systems. Representation theory investigates groups acting by linear transformations on vector spaces. Unitary irreducible representations are a noncommutative generalization of the pure harmonics seen in Fourier analysis, and they are also linked to the idea of "elementary states" in quantum mechanics. Topological groups act continuously on topological spaces and geometries - indeed Klein's Erlangen program sought to classify geometries by their symmetry. One can also see group actions e.g. in the theory of covering spaces with monodromy actions. Another application that surprised me is my reading that the Banach-Tarski paradox is heavily a fact about actions!

There is a somewhat isolated application of group actions to combinatorics: counting things modulo symmetry. This is where things like the Orbit-Stabilizer theorem or Burnside's Lemma or the full Polya's enumeration theorem come into play. (In my opinion Polya enumeration leads naturally to the theory of combinatorial species.) From what I understand this has real-world applications in chemistry in analyzing molecules, but I have no familiarity with that topic.

As with many things in math, even if the interestingness or intuitive meaning of something is difficult to comprehensively articulate and transfer from an experienced mind to another mind just beginning to become acquainted with something, time and exposure can lead to the development of an appreciation that cannot be effectively communicated on demand in words. So, do make some effort to understand what a group action is, but beyond that it might just come with time.

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A group action is really just a slight generalization of a permutation group. An equivalent definition that illustrates this is as follows:

Definition Let $G$ be a group. Let $X$ be a set. Then an action of $G$ on $X$ is homomorphism $\phi : G \rightarrow \mathrm{Sym}(X)$

In your notation, we have $g x = \phi(g)(x)$ for all $g \in G$ and $x \in X$.

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Let's examine some features of a group action that aren't immediately evident:

1) The map $x \mapsto g\cdot x$ is a bijection $X \to X$.

Proof: Suppose $g\cdot x = g\cdot y$. Then $g^{-1}\cdot(g\cdot x) = g^{-1}\cdot(g\cdot y)$,

so by (ii) we have:

$e\cdot x = (g^{-1}g)\cdot x = g^{-1}\cdot(g\cdot x) = g^{-1}\cdot(g\cdot y) = (g^{-1}g)\cdot y = e\cdot y$

and by (i) we have:

$x = e\cdot x = e\cdot y = y$, so our mapping is injective.

Given any $x \in X$, let $y = g^{-1}\cdot x$. Then:

$g\cdot y = g\cdot(g^{-1}\cdot x) = (gg^{-1})\cdot x = e\cdot x = x$, so our map is surjective.

If we call the map $x \mapsto g\cdot x$, something like $\phi_g$, we have a mapping:

$G \to \text{Sym}(X)$ given by $g \mapsto \phi_g$ from $G$ to the group of bijections on $X$.

2) The map $\phi:G \to \text{Sym}(X)$ given by $\phi(g) = \phi_g$ is a group homomorphism.

What we need to demonstrate here, is that $\phi(gh) = \phi_{gh} = \phi_g \circ \phi_h = \phi(g)\phi(h)$. We do this by testing on each and every element $x \in X$.

$\phi_{gh}(x) = (gh)\cdot x = g\cdot(h\cdot x) = g\cdot(\phi_h(x)) = \phi_g(\phi_h(x)) = (\phi_g \circ \phi_h)(x)$

3) Every group action arises in this way. Suppose we have a group homomorphism:

$\psi: G \to \text{Sym}(X)$. Convince yourself that $g\cdot x = \psi(g)(x)$ defines a group action, and that performing the steps of (2) above yields the same group homomorphism we started with.


Now let's look at an example: imagine a square in the plane, centered at the origin. We know we have a group (which we could model with $2\times2$ matrices) of order $8$, the "symmetries of the square". Typically, we take:

$r = \begin{bmatrix}0&-1\\1&0\end{bmatrix}$

$s = \begin{bmatrix}1&0\\0&-1\end{bmatrix}$

and obtain $D_4 = \{e,r,r^2,r^3,s,rs,r^2s,r^3s\}$

We can let $D_4$ act on the set of the "four corners", which gives the following subgroup of $S_4$:

$e \mapsto (1)$

$r \mapsto (1\ 2\ 3\ 4)$

$r^2 \mapsto (1\ 3)(2\ 4)$

$r^3 \mapsto (1\ 4\ 3\ 2)$

$s \mapsto (1\ 4)(2\ 3)$

$rs \mapsto (2\ 4)$

$r^2s = (1\ 2)(3\ 4)$

$r^3s = (1\ 3)$

We turned our group of $8$ matrices into a shuffling of $4$ corners. We don't get "every" shuffling, because some re-arrangements would yield a "bow-tie", and not a square. What happens if we consider instead the set of $4$ edges, or the set of $2$ diagonals?

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Let $\text{Perm}(X)$ be the permutation group of $X$, then the group action is a homomorphism $\pi:G\to\text{Perm}(X)$. Now, $(\pi(g))(x)$ can be written as $gx$.

By properties of homomorphisms, $ex=(\pi(e))(x)=\text{id}_X(x)=x$, and $ghx=(\pi(gh))(x)=(\pi(g)\circ\pi(h))(x)=g(\pi(h)(x))=g(hx)$.

I'm not sure if that helps, but it was how I made sense of group actions.

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