1. Let \(n \in \mathbb{N}.\) A permutation is a symmetry of the set \(\mathcal{X}=\{1,2,\ldots,n\}.\)

2. Let \(V\) be a \(\mathbb{K}\)-vector space and \(v_0 \in V.\) The translation \(T_{v_0} : V \to V,\) \(v\mapsto v+v_0\) by the vector \(v_0\) is a symmetry of \(V.\)

3. Let \(\mathcal{X}\) be any non-empty set. The identity transformation \(\mathrm{Id}_\mathcal{X} : \mathcal{X} \to \mathcal{X}\) defined by \(\mathrm{Id}_\mathcal{X}(x)=x\) for all \(x \in \mathcal{X}\) is a symmetry of \(\mathcal{X}.\)

1. Consider \(\mathcal{Z}=\mathbb{R}^2\) and \(\mathcal{X}\) to be the circle of radius \(r>0\) centred at the origin \(0_{\mathbb{R}^2},\) that is, \(\mathcal{X}=\{\vec{x}=(x_i)_{1\leqslant i\leqslant 2} \in \mathbb{R}^2 | (x_1)^2+(x_2)^2=r^2\}.\) Let \(\theta \in \mathbb{R}\) and \[\mathbf{R}_{\theta}=\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\] so that \(f_{\mathbf{R}_{\theta}} : \mathbb{R}^2 \to \mathbb{R}^2\) is the counter-clockwise rotation around the origin \(0_{\mathbb{R}^2}\) with angle \(\theta.\) A rotation does not change the length of a vector and hence \(f_{\mathbf{R}_{\theta}}(\vec{x}) \in \mathcal{X}\) for each element \(\vec{x} \in \mathcal{X}.\) The restriction \(\sigma=f_{\mathbf{R}_{\theta}}|_{\mathcal{X}} : \mathcal{X} \to \mathcal{X}\) of the rotation \(f_{\mathbf{R}_{\theta}}\) to the circle \(\mathcal{X}\) is thus a symmetry of the circle. Notice that not all symmetries of the circle are restrictions of rotations. The linear mapping \[f : \mathbb{R}^2 \to \mathbb{R}^2, \quad \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \mapsto \begin{pmatrix} x_1 \\ -x_2 \end{pmatrix}\] is the reflection along the \(x_1\)-axis and hence restricts to be a bijective mapping from the circle \(\mathcal{X}\) onto itself. It is thus also a symmetry of the circle.

2. Let \(n \in \mathbb{N}\) with \(n\geqslant 3.\) We consider a regular polygon \(\mathcal{X}\) with \(n\) sides centred at the origin in \(\mathcal{Z}=\mathbb{R}^2\) an so that \((1,0) \in \mathcal{X}.\) Clearly, not every rotation of \(\mathbb{R}^2\) restricts to be a symmetry of \(\mathcal{X},\) but only rotations with angle \(2\pi k/n\) where \(k \in \{0,1,2,\ldots,n-1\}.\) We thus have \(n\) rotation symmetries arising from the matrices \[\begin{pmatrix} \cos\frac{2\pi k}{n} & - \sin\frac{2\pi k}{n} \\ \sin\frac{2\pi k}{n} & \cos\frac{2\pi k}{n} \end{pmatrix}.\] In addition, the reflection along the \(x_1\)-axis is a symmetry of \(\mathcal{X}.\)

1. The symmetries of a set \(\mathcal{X}\) form a group \(G,\) often denoted by \(\mathrm{Sym}(\mathcal{X}),\) where \(*_G=\circ\) is the composition of mappings. The identity element is the identity mapping \(e_G=\mathrm{Id}_\mathcal{X}\) and the inverse of each symmetry \(\sigma\) is the mapping inverse \(\sigma^{-1}.\) In particular, for \(n \in \mathbb{N},\) the permutations of \(\mathcal{X}_n=\left\{1,2,\ldots,n\right\}\) form a group \(G=S_n\) with \(*_G=\circ\) and \(e_G=1,\) the identity permutation.

2. A field \(\mathbb{K}\) gives rise to two groups. The additive group of the field where \(G=\mathbb{K}\) and \(*_G=+_\mathbb{K}\) and the multiplicative group of the field where \(G=\mathbb{K}^*\) and \(*_G=\cdot_{\mathbb{K}}.\) For the additive group we have \(e_G=0_{\mathbb{K}}\) and the inverse of \(x\in \mathbb{K}\) is \(-x.\) For the multiplicative group we have \(e_G=1_{\mathbb{K}}\) and the inverse of \(x \in \mathbb{K}^*\) is \(\frac{1}{x}.\)

3. A \(\mathbb{K}\)-vector space \(V\) gives rise to a group where \(G=V\) and \(*_G=+_V.\) Here the identity element is the zero vector \(e_G=0_V\) and the inverse of \(v \in V\) is \(-v.\)

4. Let \(n \in \mathbb{N}.\) The invertible \(n\times n\) matrices with entries in \(\mathbb{K}\) form a group \(G\) commonly denoted by \(\mathrm{GL}_{n}(\mathbb{K})\) or \(\mathrm{GL}(n,\mathbb{K}).\) Here \(*_{G}\) is matrix multiplication, \(e_G=\mathbf{1}_{n},\) the identity matrix of size \(n\) and the inverse of a group element is the matrix inverse. \(\mathrm{GL}\) is an abbreviation of general linear.

The set of integers \(\mathbb{Z}\) is a subgroup of the Abelian group \((\mathbb{Q},+),\) where \(+\) denotes usual addition of rational numbers. Indeed \(0 \in \mathbb{Z}\) and the sum of two integers is again an integer. Recall that for \(m \in \mathbb{Z},\) the notation \(m^{-1}\) refers to the inverse element of \(m\) with respect to the group operation. So here \(m^{-1}\) is the additive inverse of \(m \in \mathbb{Z},\) that is \(-m.\) Since \(-m \in \mathbb{Z}\) for all \(m \in \mathbb{Z},\) we conclude that \(\mathbb{Z}\) is an (Abelian) subgroup of \((\mathbb{Q},+).\)

A subspace \(U\subset V\) of a \(\mathbb{K}\)-vector space \(V\) is a subgroup of the Abelian group \((V,+_V).\)

Let \(\mathrm{SL}(n,\mathbb{K})\) denote the subset of \(\mathrm{GL}(n,\mathbb{K})\) consisting of matrices of determinant \(1.\) The set \(\mathrm{SL}(n,\mathbb{K})\) is non-empty since it contains \(\mathbf{1}_{n}.\) Furthermore, the product rule for the determinant Proposition 5.21 implies that if \(\mathbf{A}, \mathbf{B}\in \mathrm{SL}(n,\mathbb{K}),\) then so is the matrix product \(\mathbf{A}\mathbf{B}.\) Corollary 5.22 furthermore implies that if \(\mathbf{A}\in \mathrm{SL}(n,\mathbb{K}),\) then so is \(\mathbf{A}^{-1}.\) It follows that \(\mathrm{SL}(n,\mathbb{K})\) – commonly also denoted by \(\mathrm{SL}_n(\mathbb{K})\) – is a subgroup of \(\mathrm{GL}(n,\mathbb{K})\) called the special linear group.

The trigonometric identities for \(\sin\) and \(\cos\) imply that \(\mathbf{R}_{\theta}\mathbf{R}_{\vartheta}=\mathbf{R}_{\theta+\vartheta},\) where \(\theta,\vartheta \in \mathbb{R}\) Since \(\mathbf{R}_0=\mathbf{1}_{2} \in \mathrm{SL}(2,\mathbb{R})\) and \(\det \mathbf{R}_\theta=1\) for all \(\theta \in \mathbb{R},\) we conclude that the rotations \(\{\mathbf{R}_{\theta} | \theta \in \mathbb{R}\}\) around the origin \(0_{\mathbb{R}^2}\) form a subgroup of \(\mathrm{SL}(2,\mathbb{R}).\) The group of rotations in \(\mathbb{R}^2\) is denoted by \(\mathrm{SO}(2).\) Later on we will encounter the orthogonal group \(\mathrm{O}(n)\) and the special orthogonal group \(\mathrm{SO}(n),\) the latter of which generalises \(\mathrm{SO}(2)\) to higher dimensions.

1. Every group \(G\) acts on itself. We take \(\mathcal{X}=G\) and define \[\phi : G \times G \to G, \quad (a,b)\mapsto \phi(a,b)=a*_G b.\] Then for all \(a \in G\) we have \(\phi(e_g,a)=e_g*_G a=a.\) Furthermore, for all \(a,b,c \in G\) we have \[\phi(a*_G b,c)=(a*_G b)*_G c=a*_G(b*_G c)=a*_G\phi(b,c)=\phi(a,\phi(b,c))\] so that \(\phi\) does indeed define an action of \(G\) on itself.

2. Consider \(\mathcal{X}=\mathbb{R}^2\) and \(G=\mathrm{SO}(2).\) We define an action \[\phi : G \times \mathcal{X} \to \mathcal{X}, \quad (\mathbf{R}_{\theta},\vec{x}) \mapsto \phi(\mathbf{R}_{\theta},\vec{x})=\mathbf{R}_{\theta}\vec{x}\] which rotates a vector \(\vec{x} \in \mathbb{R}^2\) counter-clockwise around the origin \(0_{\mathbb{R}^2}\) by the angle \(\theta.\) Here \(*_G\) is just matrix multiplication, so we have for all \(\vec{x}\in \mathbb{R}^2\) and \(\mathbf{R}_{\theta},\mathbf{R}_{\vartheta} \in \mathrm{SO}(2)\) \[\phi(\mathbf{R}_{\theta}\mathbf{R}_{\vartheta},\vec{x})=\mathbf{R}_{\theta}\mathbf{R}_{\vartheta}\vec{x}=\mathbf{R}_{\theta}\phi(\mathbf{R}_{\vartheta},\vec{x})=\phi(\mathbf{R}_{\theta},\phi(\mathbf{R}_{\vartheta},\vec{x})).\] Furthermore, since \(e_{\mathrm{SO}(2)}=\mathbf{R}_0=\mathbf{1}_{2},\) we have for all \(\vec{x} \in \mathbb{R}^2\) \[\phi(e_{\mathrm{SO}(2)},\vec{x})=\mathbf{1}_{2}\vec{x}=\vec{x}.\] It follows that \(\phi\) does indeed define an action of \(\mathrm{SO}(2)\) on \(\mathbb{R}^2.\)

3. Let \(n \in \mathbb{N}\) and \(\mathcal{X}=M_{n,n}(\mathbb{K}).\) The general linear group \(\mathrm{GL}(n,\mathbb{K})\) acts on \(\mathcal{X}\) by conjugation. We define \[\phi : \mathrm{GL}(n,\mathbb{K}) \times M_{n,n}(\mathbb{K}) \to M_{n,n}(\mathbb{K}), \quad (\mathbf{C},\mathbf{A}) \mapsto \phi(\mathbf{C},\mathbf{A})=\mathbf{C}\mathbf{A}\mathbf{C}^{-1}.\] Then for all \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) we have \[\phi(e_G,\mathbf{A})=\phi(\mathbf{1}_{n},\mathbf{A})=\mathbf{1}_{n}\mathbf{A}(\mathbf{1}_{n})^{-1}=\mathbf{A}\] where we use that \(e_{\mathrm{GL}(n,\mathbb{K})}=\mathbf{1}_{n}.\) Moreover, for \(\mathbf{C},\mathbf{C}^{\prime} \in \mathrm{GL}(n,\mathbb{K})\) and \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) we have \[\begin{aligned} \phi(\mathbf{C}\mathbf{C}^{\prime} ,\mathbf{A})&=\mathbf{C}\mathbf{C}^{\prime} \mathbf{A}(\mathbf{C}\mathbf{C}^{\prime})^{-1}=\mathbf{C}\mathbf{C}^{\prime}\mathbf{A}(\mathbf{C}^{\prime})^{-1}\mathbf{C}^{-1}\\ &=\mathbf{C}\phi(\mathbf{C}^{\prime},\mathbf{A})\mathbf{C}^{-1}=\phi(\mathbf{C},\phi(\mathbf{C}^{\prime} ,\mathbf{A})), \end{aligned}\] where we use that for all \(\mathbf{C},\mathbf{C}^{\prime} \in \mathrm{GL}(n,\mathbb{K}),\) we have \((\mathbf{C}\mathbf{C}^{\prime} )^{-1}=(\mathbf{C}^{\prime})^{-1}\mathbf{C}^{-1}.\) It follows that \(\phi\) does indeed define an action of \(\mathrm{GL}(n,\mathbb{K})\) on \(M_{n,n}(\mathbb{K}).\)

4. Let \(V\) be a \(\mathbb{K}\)-vector space and \(U\subset V\) a subspace. Taking \(G=U\) with \(*_G=+_U\) and \(\mathcal{X}=V,\) the group \(G\) acts by translation. We define \[\phi : U \times V \to V, \quad (u,v)\mapsto \phi(u,v)=u+_V v.\] Since \(e_G=0_U=0_V,\) we have for all \(v \in V\) \[\phi(e_G,v)=0_V+_V v=v.\] Moreover, for all \(u_1,u_2 \in U\) and \(v \in V\) we have \[\phi(u_1+_U u_2,v)=(u_1+_U u_2)+_V v=u_1+_V\phi(u_2,v)=\phi(u_1,\phi(u_2,v)),\] where we use that \(+_U : U \times U \to U\) is the restriction of \(+_V : V \times V \to V\) to \(U\times U\subset V\times V.\) We conclude that \(\phi\) defines an action of the subspace \(U\) on \(V.\)

5. Let \(n \in \mathbb{N}.\) A permutation \(\sigma \in S_n\) acts on \(\mathcal{X}_n=\{1,2,\ldots,n\}\) by \[\phi : S_n \times \mathcal{X}_n \to \mathcal{X}_n \quad (\sigma,m) \mapsto \phi(\sigma,m)=\sigma(m).\] We leave it to the reader to check that this is indeed an action. In addition, a permutation \(\sigma \in S_n\) does also act on \(\mathbb{K}^n\) by the rule \[\phi(\sigma,\vec{x})=\mathbf{P}_{\sigma}\vec{x},\] where \(\vec{x} \in \mathbb{K}^n\) and \(\mathbf{P}_{\sigma}\) is the permutation matrix associated to \(\sigma \in \mathbb{K}^n,\) c.f. Definition 5.28.

1. Let \(\mathcal{X}\subset \mathbb{R}^3\) denote the set of all points in our solar system. An asteroid initially at rest at the position \(x_0 \in \mathcal{X}\) will move under the influence of gravity. Let \(x_t\) denote the position of the asteroid after time \(t \in \mathbb{R}\) has passed. The mapping \[\phi : \mathbb{R}^+_{0} \times \mathcal{X} \to \mathcal{X}, \quad (t,x_0) \mapsto \phi(t,x_0)=x_t\] describing the movement of the asteroid is then a time-continuous dynamical system.

2. Let \(\mathcal{X}=\{0,1\}^N\) denote the carrier status of a contagious disease of each individual of a population of size \(N \in \mathbb{N}.\) So \(x\in \mathcal{X}\) is a list of length \(N\) containing \(0\)s and \(1\)s, where the \(k\)-th entry reflects the carrier status of the \(k\)-th member of the population, \(0\) for non-carriers and \(1\) for carriers. Let \(x_0 \in \mathcal{X}\) denote the carrier status at some initial time \(t=0\) and for \(m \in \mathbb{N}_{0}\) let \(x_m\) denote the carrier status after \(m\) days have passed. The mapping \[\phi : \mathbb{N}_{0} \times \mathcal{X} \to \mathcal{X}, \quad (m,x_0) \mapsto \phi(m,x_0)=x_m\] describing the progression of the disease in the population is then a time-discrete dynamical system.

1. Consider the action of \(\mathrm{SO}(2)\) on \(\mathbb{R}^2\) from above. The orbit of \(\vec{x}\neq 0_{\mathbb{R}^2}\) consists of all points in \(\mathbb{R}^2\) obtained by rotating \(\vec{x}\) counter-clockwise around the origin. Since the rotation angle can be chosen arbitrarily, the orbit of \(\vec{x}\) is the circle of all points of \(\mathbb{R}^2\) that have the same length as \(\vec{x}.\) On the other hand, the orbit of \(0_{\mathbb{R}^2}\) only consists of \(0_{\mathbb{R}^2},\) that is, we have \[\mathrm{SO}(2)*_{\mathrm{SO}(2)}0_{\mathbb{R}^2}=\{0_{\mathbb{R}^2}\}.\] In this particular case we have a complete picture of all possible orbits, an orbit is either the zero vector or else a circle centred at the origin, hence \[\mathcal{X}/G=\mathbb{R}^2/\mathrm{SO}(2)=\{0_{\mathbb{R}^2}\}\cup\{\text{circle of radius }r \text{ centred at }0_{\mathbb{R}^2} | r>0\}.\]

2. Consider the action of \(\mathrm{GL}(n,\mathbb{K})\) on \(M_{n,n}(\mathbb{K})\) from above. Let \(\mathbf{D}=\mathrm{diag}(\lambda_1,\ldots,\lambda_n)\) be a diagonal matrix with entries \(\lambda_1,\ldots,\lambda_n \in \mathbb{K}.\) The \(\mathrm{GL}(n,\mathbb{K})\)-orbit of \(\mathbf{D}\) then consists of all \(n\times n\)-matrices with entries in \(\mathbb{K}\) that are diagonalisable with eigenvalues \(\lambda_1,\ldots,\lambda_n.\) A complete description of the set of orbits \(M_{n,n}(\mathbb{K})/\mathrm{GL}(n,\mathbb{K})\) is out of reach for us at this point, we will however have more to say about this in the Linear Algebra II module.

3. We consider the action of \(S_2\) on \(\mathbb{R}^2\) as defined above. The orbit of a vector \(\vec{v}=\begin{pmatrix} x \\ y \end{pmatrix}\in\mathbb{R}^2\) with \(x\neq y\) is the subset \[S_2*_{S_2}\vec{v}=\left\{\begin{pmatrix} x \\ y \end{pmatrix},\begin{pmatrix} y \\ x \end{pmatrix}\right\}.\] On the other hand, the orbit of a vector \(\vec{v}=\begin{pmatrix} x \\ x \end{pmatrix}\in \mathbb{R}^2\) is just \(\{\vec{v}\}.\) For a vector of the first type, either \(x>y\) or \(x<y.\) The orbit of each such vector can thus be represented uniquely by a vector \(\vec{v}=\begin{pmatrix} x \\ y \end{pmatrix}\) with \(y>x.\) The vectors of the second type lie on the axis defined by the equation \(y=x.\) We thus have a bijective mapping from \(\mathbb{R}^2/S_2\) to the half plane \(\mathbb{H}=\{(x,y) \in \mathbb{R}^2 | y\geqslant x\}\).