Monoids of Functions and Matrices, Permutation Groups, and the Matrix Exponential as a One-Parameter Group

In the settings of Groups: Standing Notation and Euclidean Space and Real Matrices: Standing Notation (Algebra), let $X$ be a set and let $n\in\mathbb{N}$.

(i) Functions of a set. The set $X^{X}$ of functions from $X$ to $X$, with composition, is a monoid with identity $\mathrm{id}_{X}$.

(ii) Units of the function monoid. The units of the monoid $X^{X}$ of (i) are exactly the permutations of $X$.

(iii) The group of permutations of a set. The permutations of $X$, with composition, form a group.

(iv) Symmetric groups. The set $S_{n}$ of permutations of $[n]$, with composition, is a group.

(v) The signs. $\{1,-1\}$ is a subgroup of the multiplicative group $\mathbb{R}\setminus\{0\}$ of Additive and Multiplicative Groups of Numbers (ii).

(vi) The sign homomorphism. The sign $\operatorname{sgn}:S_{n}\to\{1,-1\}$ is a homomorphism from the group $S_{n}$ of (iv) to $\{1,-1\}$ with multiplication, a subgroup by (v):

$$\operatorname{sgn}(\sigma\circ\tau)=\operatorname{sgn}(\sigma)\operatorname{sgn}(\tau)\qquad(\sigma,\tau\in S_{n}).$$

(vii) Square matrices. $\mathcal{M}_{n}(\mathbb{R})$, with the matrix product, is a monoid with identity $I_{n}$.

(viii) Units of the matrix monoid. The units of the monoid $\mathcal{M}_{n}(\mathbb{R})$ of (vii) are exactly the invertible matrices.

(ix) The general linear group. $\mathrm{GL}_{n}(\mathbb{R})$, with the matrix product, is a group.

(x) The determinant. $\det:\mathrm{GL}_{n}(\mathbb{R})\to\mathbb{R}\setminus\{0\}$ is a homomorphism from the group $\mathrm{GL}_{n}(\mathbb{R})$ of (ix) to the multiplicative group $\mathbb{R}\setminus\{0\}$ of Additive and Multiplicative Groups of Numbers (ii):

$$\det(AB)=\det A\,\det B\qquad(A,B\in\mathrm{GL}_{n}(\mathbb{R})).$$

(xi) The matrix exponential. For every $A\in\mathcal{M}_{n}(\mathbb{R})$, the function $\mathbb{R}\to\mathrm{GL}_{n}(\mathbb{R})$, $t\mapsto e^{tA}$, with $e^{M}$ the exponential of $M\in\mathcal{M}_{n}(\mathbb{R})$, is a one-parameter group in the group $\mathrm{GL}_{n}(\mathbb{R})$ of (ix).