MATH 415 Lecture 9

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First test will be Oct. 8

Generating Sets

Given a group $G$ and $X\subset G$ is a generating set for $G$ if the minimum subgroup of $G$ containing $X$ is $G$

if $H\leq G$ and $X\subset H$, then $H=G$.

For all $g\in G$, we can represent $g$ as a product of elements of $X$ to power $\pm 1$:

$g=x_{1}^{\epsilon _{1}}\,\dots \,x_{n}^{\epsilon _{n}}$ for $x\in X$ and $\epsilon _{i}\in \left\{-1,1\right\}$

We denote $G=\left\langle X\right\rangle =\left\{x_{1}^{\epsilon _{1}}\,\dots \,x_{n}^{\epsilon _{n}}~\mid ~x\in X,\ \epsilon _{i}\in \left\{-1,1\right\}\right\}$ as the subgroup generated by the generating set $X$.

$G$ is finitely generated if there is a finite generating set for $G$.

Examples

• $\mathbb {Z}$ is finitely generated by $X=\left\{1\right\}$ or by $X=\left\{2,3\right\}$.
• $\mathbb {Z} \oplus \mathbb {Z}$ is also finitely generated: $X=\left\{(0,1),(1,0)\right\}$
• $\left\langle \mathbb {Q} ,+\right\rangle$ is not finitely generated: If $\mathbb {Q}$ is finitely generated, then $|X|<\infty$ and $X\subset \mathbb {Q}$, then $\mathbb {Q} =\left\langle X\right\rangle$ must be cyclic, which it is not. Contradiction.

Fundamental Theorem of Finitely Generated Abelian Groups

Theorem 11.12

The fundamental theorem for finitely generated abelian groups

Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of the form

$\mathbb {Z} _{p_{1}^{r_{1}}}\times \mathbb {Z} _{p_{2}^{r_{2}}}\times \dots \times \mathbb {Z} _{p_{n}^{r_{n}}}\times \mathbb {Z} \times \dots \times \mathbb {Z} \,\!$

Where $p_{i}$ are prime numbers, and $r_{i}\in \mathbb {N}$

This representation is unique up to the rearrangement of factors.

Recall that all cyclic groups are isomorphic to $\mathbb {Z}$ if infinite and $\mathbb {Z} _{n}$ if finite and of order $n$.

Also recall that $\mathbb {Z} _{m}\times \mathbb {Z} _{n}\simeq \mathbb {Z} _{m\,n}$ if $m$ and $n$ are coprime. All primes are coprime to other primes

The (number of)^? infinite factors $\beta (G)$ is an invariant of $G$ and is called the Betti number.

The (number of)^? prime groups is also an invariant.

Example

Let $G$ be an abelian group with $\left|G\right|=360=2^{3}\cdot 3^{2}\cdot 5$. What is $G$?

1. $\mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{3}\times \mathbb {Z} _{3}\times \mathbb {Z} _{5}$
2. $\mathbb {Z} _{2}\times \mathbb {Z} _{4}\times \mathbb {Z} _{3}\times \mathbb {Z} _{3}\times \mathbb {Z} _{5}$
3. $\mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{9}\times \mathbb {Z} _{5}$
4. $\mathbb {Z} _{2}\times \mathbb {Z} _{4}\times \mathbb {Z} _{9}\times \mathbb {Z} _{5}$
5. $\mathbb {Z} _{8}\times \mathbb {Z} _{3}\times \mathbb {Z} _{3}\times \mathbb {Z} _{5}$
6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_5}

This is an exhaustive list (by theorem 11.12) of all abelian groups that contain 360 elements. Any other groups can be decomposed into above: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_{360} \simeq \mathbb{Z}_{40} \times \mathbb{Z}_9 \simeq \mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_5} .

Decomposable Groups

A group Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is decomposable if $G\simeq H\times K$ with $\left|H\right|\neq 1$ and $\left|K\right|\neq 1$.

Theorem 11.15

The finite indecomposible abelian groups are exactly the cyclic groups whose order is a power of a prime: $\mathbb {Z} _{p}^{r}$.

Proof. Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is indecomposible. Apply the fundamental theorem to arrive at the conclusion that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \simeq \mathbb{Z}_{p_i^{r_i}}} .

Conversely, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \mathbb{Z}_{p^r}} . Assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \mathbb{Z}_{p^r} = \mathbb{Z}_{p_i} \times \mathbb{Z}_{p_j}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = i + j} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i > j} . We know that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_{p^r}} is cyclic, thus it has a generator of order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^r} . If we take an element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g = (a,b) \in \mathbb{Z}_{p^i} \times \mathbb{Z}_{p^j}} , then the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is the least common multiple of the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in \mathbb{Z}_{p^i}} and the order of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \in \mathbb{Z}_{p^j}} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{order}(g) \le \lcm{(p^i, p^j)}} Since we have the order of an element is at most Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^i} for the subgroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_{p^i}} , this contradicts the fact that the same element in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \mathbb{Z}_{p^r}} must have an order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^r} because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^i < p^r} .

quod erat demonstrandum

Theorem 11.16

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \mid \left| G \right|} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a finite abelian group, then there is a subgroup Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \le G} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| H \right| = m} .

Homomorphisms

A map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi : G \to G'} is called a homomorphism if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x\,y) = \phi(x) \, \phi(y)} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y \in G} .

1. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is injective (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \ne y} implies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) \ne \phi(y)} ), then it is called a monomorphism
2. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is surjective (for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in G'} there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in G} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = y} ), then it is called an epimorphism

The trivial homomorphism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = e} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} is the identity element in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G'} , is simple but not interesting.

Theorem

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi : G \to G'} is surjective and $G$ is abelian, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G'} is also abelian.

Proof. Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y,z \in G'} . Is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\,z = z\,y} ?

Because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is onto, we know that there exist Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u,v \in G} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(u) = y} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(v) = z} . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is abelian, we know $u\,v=v\,u$. By applying our homomorphism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} , we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(u\,v) &= \phi(u) \, \phi(v) = y \, z \\ \phi(v\,u) &= \phi(v) \, \phi(u) = z \, y }

Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G'} is abelian by the above result.

quod erat demonstrandum

Examples

Symmentric Group of Permutations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n} is a symmetric group. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi : S_n \to \mathbb{Z}_2} with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(\sigma) = \begin{cases} 0 & \sigma \ \mbox{is an even permutation} \\ 1 & \mbox{otherwise} \end{cases}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is a homomorphism.

Note that $S_{n}$ is not commutative, but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_2} is commutative.

Evaluation Homomorphism

Define the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = \left\{ f : \mathbb{R} \to \mathbb{R} \right\}} as a function over real numbers. We claim Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle F, + \right\rangle} is an abelian group.

Indeed, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) + g(x) = g(x) + f(x)} .

We can represent each function as a graph on cartesian plane. Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_c : F \to \mathbb{R}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} is the additive group of real numbers, as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_c(f) = f(c) \in \mathbb{R}}

Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_c(f+g) = (f+g)(c) = f(c) + g(c)} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is a homomorphism

Example: Vector Addition

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} be the additive group of column vectors. then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x} = \left\langle x_1, \ldots, x_n \right\rangle \in \mathbb{R}^n} . We define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_A : \mathbb{R}^n \to \mathbb{R}^m} to be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_A (\vec{x}) = A \, \vec{x}}

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} matrix.

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_A(\vec{x} + \vec{y}) = A(\vec{x} + \vec{y}) = A\,\vec{x} + A\,\vec{y} = \phi_A(\vec{x}) + \phi_A(\vec{y})} . Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is a homomorphism.

Example: Determinants

Recall that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det{A\,B} = \det{(A)} \, \det{(B)}} . Then the determinant is a homomorphism:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det : M_{n}^*(\mathbb{R}) \to \mathbb{R}^*} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^*} is the multiplicative group over nonzero real numbers.

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{n}^*(\mathbb{R})} is not commutative, but $\mathbb {R} ^{*}$ is commutative.