Section 4 Groups

Group is a algebra structure:

Algebra Structure
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Definition: Group - normal definition

A group $⟨ G, * ⟩$ is a set $G$ , with a operation $*$ , such that the following axioms are satisfied:

Algebra Chapter 0, p.63

Definition: A group is a groupoid with a single object.

We consider this groupoid G:

objects: ${*}$
Morphisms: This single object has only endomorphisms (an object to itself). Moreover, they are automorphisms, because morphisms in groupoids are isomorphisms. (see Morphisms > Definition Automorphism). Thus the only hom-set is $H o m_{G} (*, *)$ or $A u t_{G} (*)$

NOTE

The set $A u t_{G} (*)$ and the morphism composition forms a group.

We check the criterions:

all morphisms are invertible (because of isomorphism)
for $f, g \in A u t_{G} (*)$ , $g \circ f$ is again an automorphism of G
By Definition of category, the morphism composition fulfills the associativity law. (see Definition of Category)
By Definition of category, there is an identity automorphism. (see Definition of Category)

Thus, we can also say

Important

The hom-set of a single-object groupoid forms a group under composition.

We can conclude that: $G r p ≅ Gp d_{*}$

A groupoid G defined as $({*}, H o m_{G} (*, *) = Z)$ : The morphisms in this set are $f_{n} : * \to *$

$f_{n}$ here is purely a symbol, representing a morphism. We define a composition: $\circ : H o m_{G} (*, *) \times H o m_{G} (*, *) \to H o m_{G} (*, *)$
$\circ$ is defined as: For any $m, n \in Z$ : $f_{n} \circ f_{m} = f_{n + m}$ $i d_{*} = f_{0}$ and the inverse is defined as $(f_{n})^{- 1} = f_{- n}$