Group

A group is a fundamental algebraical structure. Groups arise in geometry and physics as symmetries of spaces.

Definition

First properties

Definition Group A group is a set $𝑋$ equipped with a binary operation $- \cdot - : 𝑋^{2} \to 𝑋$ , such that

Associativity: $\cdot$ is associative.
Unitality: There exists a neutral element $1 : 𝑋$ for $\cdot$
Inversability: For every $𝑥 : 𝑋$ , there exists an inverse $𝑥^{- 1} : 𝑋$ such that
$𝑥 \cdot 𝑥^{- 1} = 𝑥^{- 1} \cdot 𝑥 = 1$

Lemma Let

𝐺

be a group. The neutral element, and the inverses, are unique.

Proof.

Uniqueness of neutral: Let $𝑒$ and $𝑒^{'}$ be two neutral elements. We have
$𝑒 = 𝑒 \cdot 𝑒^{'} = 𝑒^{'}$
Uniqueness of inverses: Let $𝑥 : 𝐺$ , and $𝑦$ and $𝑧$ be two inverses of $𝑥$ . We have
$\begin{aligned} 𝑦 & = 𝑦 \cdot 𝑒 \\ = 𝑦 \cdot 𝑥 \cdot 𝑧 \\ = 𝑧 \end{aligned}$

Groups of automorphisms

Let $𝒞︀$ be a previewcategory, and $𝑋 : 𝒞︀$ be an object. We can form its group of previewautomorphisms $Aut (𝑋)$ as follows: elements of $Aut (𝑋)$ are isomorphisms $𝑓 : 𝑋 \to 𝑋$ in $𝒞︀$ , with inner operation the composition. Groups of automorphisms are extremely common.

Example Symmetry group Let $𝑋$ be a set. We note $𝔖_{𝑋}$ the group of its bijections, called permutations.

A sub-group of $𝔖_{𝑋}$ is called a permutation group.

We note

𝔖_{𝑛} ≔ 𝔖_{⟦ 1, 𝑛 ⟧}

the canonical group of symmetries over a finite set of cardinal.

Example Linear group Let

𝑉

be a linear space. We note

GL (𝑉)

Order of a group

The cardinal of a group is so important in caracterizing groups that it has its own name:

Definition Group order Let

𝐺

be a group. We note its cardinal

| 𝐺 |

, and say it is its order.

Those two examples are instances of groups of automorphisms in a given category, respectively for $𝐒𝐞𝐭$ and $𝐕𝐞𝐜𝐭$ .

Definition Group product Let

{(𝐺_{𝑖})}_{𝑖 : 𝐼}

be a family of groups. We can canonically equip

\prod_{𝑖 : 𝐼} 𝐺_{𝑖}

with a group law00 Given by pointwise product., which makes it the product group of the $𝐺_{𝑖}$ .

This group is called the product of the $𝐺_{𝑖}$ because it is a cartesian product in the category $𝐆𝐫𝐩$ .

Definition Abelian group We say that a group

𝐺

is abelian if its multiplication is commutative

Example

ℤ

ℚ

ℝ

and

ℂ

are abelian groups (for

+

). Similarly,

{\pm 1}

ℚ^{*}

ℝ^{*}

ℂ^{*}

𝕌_{𝑛}

𝕌_{\infty}

𝕌 ≔ 𝕊^{1}

are abelians (for

\times

Example There exists a unique group law over

ℤ / (𝑛 ℤ)

such that

𝜋 : ℤ \to ℤ / (𝑛 ℤ)

is a group morphism.

Definition Sub-group A sub-group of a group

𝐺

is a group

𝐻 \subseteq 𝐺

such that the inlcusion is a group morphism.

Example

$ℤ \leq ℚ \leq ℝ \leq ℂ$
$GL (𝑉) \leq 𝔖_{𝑉}$
Let $𝐸$ be a Euclidian space. An isometry is a bijection of $𝐸$ that preserves the metric of $𝐸$ . We note the group is isometries $𝐼_{𝑠} (𝐸)$ . It holds that
$𝐼_{𝑠} (𝐸) \leq$ frak𝐸
$𝑂 (𝐸) \leq 𝐼_{𝑠} (𝐸)$ , where $𝑂 (𝐸)$ is the orthogonal group. As soon as an isometry fixes $𝑂 (𝐸)$ , it is linear.

Definition Generated sub-group Let

𝐺

be a group, and

𝑋 \subseteq 𝐺

a subset of

𝐺

. The sub-group

⟨ 𝑋 ⟩ \leq 𝐺

generated by

𝑋

is the smallest sub-group of

𝐺

that contains

𝑋

One says that $𝑋$ generates $𝐺$ if $⟨ 𝑋 ⟩ = 𝐺$ .