Tag Archives: Group action

Understanding the concept of group action via an example

The following self-contained post clarifies the definition of group action (definition 1.2, p. 3, in the book “Groups, graphs and trees: an introduction to the geometry of infinite groups” by John H. Meier) by adapting an example from chapter 7, p. 72-74, of the book “A book of abstract algebra” by Charles C. Pinter.

A rigorous definition of group action and of its group homomorphism representation was provided in a previous post. This post presents the group action on the symmetries of the square to exemplify the concept.

Consider a square with numbered vertices, as shown in the figure below. The set of vertices of the square is X=\left\{1,2,3,4\right\}.

square_vertices

A symmetry of the square can be informally thought of as a way of moving the square so that it coincides with its former position. Every such move is fully described by its effect on the vertices, in the sense that every new vertex position coincides with a distinct former vertex position.

There are exactly 8 symmetric moves of the square, each of which can be described by a permutation of the square’s vertices. Here is the list of the 8 symmetries of the square:

  • The identity R_0=\left(\begin{matrix}1 & 2 & 3 & 4\\  1 & 2 & 3 & 4 \end{matrix}\right), which does not move the square.
  • Clockwise rotation of the square about its centre P by an angle of 90^{\circ}R_1=\left(\begin{matrix}1 & 2 & 3 & 4\\  2 & 3 & 4 & 1\end{matrix}\right).
  • Clockwise rotation of the square about its centre P by an angle of 180^{\circ}R_2=\left(\begin{matrix}1 & 2 & 3 & 4\\  3 & 4 & 1 & 2\end{matrix}\right).
  • Clockwise rotation of the square about its centre P by an angle of 270^{\circ}R_3=\left(\begin{matrix}1 & 2 & 3 & 4\\  4 & 1 & 2 & 3\end{matrix}\right).
  • Flip of the square about its diagonal A (see figure below): R_4=\left(\begin{matrix}1 & 2 & 3 & 4\\  1 & 4 & 3 & 2\end{matrix}\right).
  • Flip of the square about its diagonal CR_5=\left(\begin{matrix}1 & 2 & 3 & 4\\  3 & 2 & 1 & 4\end{matrix}\right).
  • Flip of the square about its vertical axis BR_6=\left(\begin{matrix}1 & 2 & 3 & 4\\  2 & 1 & 4 & 3\end{matrix}\right).
  • Flip of the square about its horizontal axis DR_7=\left(\begin{matrix}1 & 2 & 3 & 4\\  4 & 3 & 2 & 1\end{matrix}\right).

square_dihedral_group

The set of all 8 symmetries of the square is denoted by D_4=\left\{R_0,R_1,R_2,R_3,R_4,R_5,R_6,R_7\right\}. Define the operation \circ : D_4\times D_4\rightarrow D_4 to be the function composition R_i\circ R_j(x)=R_i(R_j(x)) for x\in X\left\{1,2,3,4\right\}. For example,

R_1\circ R_6 is the result of first flipping the square about its vertical axis B and then rotating it clockwise by 90^{\circ}:

R_1\circ R_6 =\left(\begin{matrix}1 & 2 & 3 & 4\\  2 & 3 & 4 & 1\end{matrix}\right)\circ\left(\begin{matrix}1 & 2 & 3 & 4\\  2 & 1 & 4 & 3\end{matrix}\right)=\left(\begin{matrix}1 & 2 & 3 & 4\\  3 & 2 & 1 & 4\end{matrix}\right)=R_5.

R_1\circ R_6 =R_5 means that first flipping the square about its vertical axis B and then rotating it clockwise by 90^{\circ} is the same as flipping the square about its diagonal C.

The set of symmetric permutations D_4 of the square along with the operation of permutation composition induces the so-called dihedral group (D_4, \circ) of the symmetries of the square. |D_4| denotes the order of group (D_4, \circ), which is the number of elements of D_4. Obviously, |D_4|=8.

The symmetric group (\mbox{Sym}(X), \circ) of X=\left\{1,2,3,4\right\} is the set of all the permutations of X, i.e. the set of all the bijective functions from X to X. Since \mbox{Sym}(X) has 4! elements, the order of \mbox{Sym}(X) is |\mbox{Sym}(X)|=24.

Note that D_4\subseteq \mbox{Sym}(X) and that (D_4, \circ) is a subgroup of (\mbox{Sym}(X), \circ).

Any function \phi:D_4\times X\rightarrow X is a group action as long as it satisfies

  • \phi(R_0, x)=x for all x \in X=\left\{1,2,3,4\right\} (identity property) and
  • \phi(R_i\circ R_j, x)=\phi(R_i, \phi(R_j, x)) for all i, j=0,1,2,3,4,5,6,7 and for all x \in X=\left\{1,2,3,4\right\} (compatibility property).

One way of picking a specific group action \phi relies on defining the type of associated group homomorphism h:D_4\rightarrow\mbox{Sym}(X) in a way that respects the identity and compatibility properties of \phi:D_4\times X\rightarrow X.

The simplest possible example would be to choose the group homomorphism h:D_4\rightarrow\mbox{Sym}(X) to be the identity function h(R_i)=R_i,~i=0,1,2,3,4,5,6,7, in which case the group action \phi takes the form \phi(R_i, x)=R_i(x).

It is easy to check that the the group action \phi(R_i, x)=R_i(x), which arises by setting the group homomorphism h to be the identity function, satisfies the properties of identity and compatibility:

  • \phi(R_0, x)=R_0(x)=x,
  • \phi(R_i\circ R_j, x)=R_i\circ R_j(x)=R_i(R_j(x))=R_i(\phi(R_j,x))=\phi (R_i,\phi (R_j, x)).

It is also easy to see for instance that the group action \phi(R_i, x)=R_i(x) maps (R_1, 2)\in D_4\times X to 3\in X, since \phi(R_1, 2)=R_1(2)=3.

The group action \phi(R_i, x)=R_i(x) is interpreted as the function that maps every symmetric move (permutation) R_i of the square and every square vertex x to the square vertex R_i(x).

The group homomorphism h(R_i)=R_i is the identity, so it is an injective function. As elaborated in this previous post, since h with h(R_i)=R_i is injective, the group action \phi with \phi(R_i, x)=R_i(x) is faithful.

A note on faithful group actions

The following self-contained post clarifies the definition of faithful group action (definition 1.2, p. 3, in the book “Groups, graphs and trees: an introduction to the geometry of infinite groups” by John H. Meier).

Let G be a group, X a set and \mbox{Sym}(X) the symmetric group of X. As it is known and as it has been elaborated in a previous post, a function \phi : G \times X \rightarrow X is a group action if and only if the function h:G\rightarrow \mbox{Sym}(X),~h(g)=f_g, with f_g(x)=\phi(g, x)=:gx, is a group homomorphism. The purpose of the present post is to clarify the concept of faithful group action.

It is easy to show that (\forall g\in G, g\ne e )(\exists x\in X)gx\ne x if and only if (\forall g,s\in G, g\ne s )(\exists x\in X)gx\ne sx. Either of these two equivalent statements defines a faithful group action \phi.

The faithful property of a group action \phi is equivalent to properties of the associated group homomorphism h. More concretely, it is easy to show that a group action \phi is faithful if and only if the associated group homomorphism h is injective if and only if h has a trivial kernel. Shortly, \mbox{Ker}(h)=\left\{e\right\}\Leftrightarrow \phi~\mbox{faithful}\Leftrightarrow h~\mbox{injective}, where \mbox{Ker}(h) is the kernel of h and e the neutral element of G.

It can also be shown that if \phi is injective, then \phi is faithful. However, the converse does not hold. So, the well-known equivalence between injective and faithful actions states that a group action \phi is faithful if and only if its associated group homomorphism h is injective (whereas \phi~\mbox{faithful}\Leftrightarrow\phi~\mbox{injective} does not hold).

Group actions as group homomorphisms

The following self-contained post relates to definition 1.2, p. 3, of the book “Groups, graphs and trees: an introduction to the geometry of infinite groups” by John H. Meier.

I have found the interpretation of the definition of group action as a group homomorphism to be verbose. This blog post provides a formalistic exposition of the equivalence between a group action and its induced homomorphism.

Let G be a group and X a set. A group action \phi of G on X is a function \phi : G \times X \rightarrow X that satisfies the properties of

  • identity, i.e (\forall x\in X)\phi(e, x)=x, where e is the identity element of G,
  • and compatibility, i.e. (\forall g, s\in G)(\forall x\in X)\phi(gs, x)=\phi(g, \phi(s, x)).

For convenience, the shortands ex=x and (gs)x=g(sx) are used in place of \phi(e, x)=x and \phi(gs, x)=\phi(g, \phi(s, x)), respectively.

Let \mbox{Sym}(X) denote the symmetric group of X, that is the group of bijective functions from X to X. Consider the family of bijective functions \left\{f_g:g\in G\right\}\subseteq\mbox{Sym}(X), where f_g:X\rightarrow X and f_g(x)=\phi(g, x)=gx.

It is now possible to define the function h:G\rightarrow \mbox{Sym}(X) as h(g)=f_g. In words, h maps an element g\in G to the bijective function f_g:X\rightarrow X defined by f_g(x)=\phi(g, x)=gx.

The common representation of a group action as a homomorphism relies on the following equivalence; a function \phi : G \times X \rightarrow X is a group action if and only if the function h:G\rightarrow \mbox{Sym}(X),~h(g)=f_g, with f_g(x)=\phi(g, x)=gx, is a group homomorphism. The main point of this blog post has been to formalize this statement. For the sake of completeness, its proof will be provided, albeit being simple.

Firstly assume that \phi is a group action. It will be shown that h is a homomorphism. By definition, h(gs)=f_{gs}. Moreover, h(g)h(s)=f_gf_s, where the product f_gf_s denotes function composition, the operation of the symmetric group \mbox{Sym}(X). Since \phi is a group action, it follows that (\forall x\in X) f_{gs}(x)=(gs)x=g(sx)=(f_gf_s)(x), therefore h(gs)=h(g)h(s), so h is a group homomorphism.

Conversely, assume that h is a homomorphism. It will be shown that \phi is a group action.

The identity e\in G belongs to the kernel of h; h(e)=h(ee), whence h(e)=h(e)h(e), so h(e)=\varepsilon, with \varepsilon(x)=x being the identity function in \mbox{Sym}(X). Furthermore, h(e)=f_e, so f_e=\varepsilon, which means that (\forall x\in X)ex=f_e(x)=\varepsilon(x)=x. The property of identity is thus satisfied for \phi.

Since h is a homomorphism, it follows that h(gs)=h(g)h(s), so f_{gs}=f_g f_s. Finally, (\forall x\in X)(gs)x=f_{gs}(x)=(f_gf_s)(x)=g(sx), hence the compatibility property of \phi is also satisfied, and \phi is a group action.