# What is the intuition behind the definition of the action of a group on a set? Why do we call a homomorphism $G\to\mathcal S(X)$ a “representation”?

Let $$G$$ be a group acting on a set $$X$$. For all $$g\in G$$, we consider the application \begin{align*} \varphi_{g}: &~ X \longrightarrow X \\ &~ x \longrightarrow g.x \end{align*} It is clear that $$\varphi_{gh}=\varphi_g\circ\varphi_h$$, $$\forall~ g,h \in G,$$ $$\varphi_e=\text{Id}_X$$ ( $$e$$ the neutral element of $$G$$ ), and $$\varphi_g \circ \varphi_{g^{-1}}=\varphi_{g^{-1}} \circ \varphi_g$$, so $$\varphi_g$$ is bijective, for all $$g\in G$$. i.e: $$\varphi \in \mathcal S(X)$$, $$\forall~ g\in G$$, and the application: \begin{align*}\Phi :&~ G \longrightarrow \mathcal S(X) \\ &~g\longrightarrow \varphi_g\end{align*} is a group homomorphism that we call a representation of $$G$$ in $$\mathcal S(X).$$

I didn't understand what this definition is for, I'm looking for the intuition or the idea behind it. Why do we call the application $$\Phi$$ by this name: "representation", it is only a homomorphism? if anyone has any ideas or comments that he can add, I will be very grateful.

• en.wikipedia.org/wiki/Group_representation may give you more information on the term. – freakish Aug 11 at 9:53
• Explore the action of $D_8$ (dihedral group on 4 vertices) on the vertex set $V = \{ 1, 2, 3, 4\}$. Write down the group table by hand. This will give you a good intuition. – abcd123 Aug 11 at 9:56
• The original purpose of groups was to describe symmetry as a collection of transformations. It was later abstracted into, well, the definition of an abstract group by a bunch of axioms that makes no reference to transformations. As a result, an abstract group can be represented by transformations of various other mathematical objects (or even other things besides transformations, like numbers or loops) in more than one way. For instance, $S_4$ could be represented by permutations of $4$ objects, or it could be represented by 3D rotations that preserve a cube. – runway44 Aug 11 at 9:57
• The definition of a group as we know it goes back to 1882 (von Dyck). Sylow's theorem was proved in 1872. – David A. Craven Aug 11 at 10:42
• See also the discussion here. – Arturo Magidin Aug 11 at 18:49

It's quite hard to understand dihedral groups abstractly, just using muliplication tables, but easy to understand them as symmetries of an $$n$$-gon via a matrix representation (i.e., as $$2\times 2$$ matrices over $$\mathbb{R}$$), or as permutations of the vertices of the $$n$$-gon.
• Complex conjugation does not affect the equation $ax^3+bx^2+cx+d=0$ (if $a,b,c,d\in\mathbb{R}$), so cannot send roots of the equation to non-roots. Therefore it permutes them. – David A. Craven Aug 11 at 10:44