Section 4 Groups

Group is a algebra structure:

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Text Elements

Monoid

Semigroup

Group

Field

• identity element

• inverses

Abelian Group

• commutativity

• second operation (multiplication)

• commutative multiplication

Division Ring (Skew Field)

Commutative Ring

• commutative multiplication

• invertibility (non-zero)

Ring

• multiplicative identity

Magma

Set

• binary operation

• associativity

Rng (without Unity)

Semiring

• additive inverses

• associative& distributive

• second operation (multiplication)

• associative& distributive

• invertibility (non-zero)

Integral Domain

• no zero-divisors

• invertibility (non-zero)

Commutative Monoid

• commutativity

• the absorption property of zero

Link to original

Definition: Group - normal definition

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

• The set is closed under $\ast$
• associativity of $\ast$
• identity element e for $\ast$
• Every element in $G$ is invertible

Definition: Group - from the view of Category Theory

Algebra Chapter 0, p.63

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

see Definition of Groupoid

We consider this groupoid G:

• objects: $\{\ast \}$
• Morphisms: This single object has only endomorphisms (an object to itself). Moreover, they are automorphisms, because morphisms in groupoids are isomorphisms. (see Definition Automorphism). Thus the only hom-set is $Ho{m}_G(\ast ,\ast )$ or $Au{t}_G(\ast )$

NOTE

The set $Au{t}_G(\ast )$ and the morphism composition forms a group.

We check the criterions:

• all morphisms are invertible (because of isomorphism)
• for $f,g\in Au{t}_G(\ast )$, $g\circ f$ 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: $Grp\cong Gp{d}_{\ast }$

Example: $⟨\mathbb{Z},+⟩$

A groupoid G defined as $(\{\ast \},Ho{m}_G(\ast ,\ast )=\mathbb{Z})$: The morphisms in this set are $f_n:\ast \to \ast$

• $f_n$ here is purely a symbol, representing a morphism. We define a composition: $\circ :Ho{m}_G(\ast ,\ast )\times Ho{m}_G(\ast ,\ast )\to Ho{m}_G(\ast ,\ast )$
• $\circ$ is defined as: For any $m,n\in \mathbb{Z}$: $f_n\circ f_m=f_{n+m}$ $i{d}_{\ast }=f_0$ and the inverse is defined as $(f_n{)}^{-1}=f_{-n}$