# Group

## Definition

An algebraic structure $$(G, \circ)$$ is a group if and only if:

 (G0) Closure $$\forall x,y \in G : x \circ y \in G$$ (G1) Associativity $$\forall x,y,z \in G : (x \circ y) \circ z = x \circ (y \circ z)$$ (G2) Identity $$\exists 1 \in RE : \forall x \in G : 1 \circ x = x = x \circ 1$$ (G3) Inverse $$\forall x \in G : \exists x^{-1} \in G : x \circ x^{-1} = 1 = x^{-1} \circ x$$

## Axioms

        assoc: ?a * ?b * ?c = ?a * (?b * ?c)
left_neutral: 1 * ?a = ?a
right_neutral: ?a * 1 = ?a
left_inverse: inverse ?a * ?a = 1
right_inverse: ?a * inverse ?a = 1

THEOREM

The group inverse is an involution.

$$\left(g^{-1}\right)^{-1} = g$$

Proof available

THEOREM    $$\forall a,b,x \in G : ax = b \Leftrightarrow x = a^{-1}b$$ $$\forall a,b,x \in G : xa = b \Leftrightarrow x = ba^{-1}$$

Proof available

THEOREM

Let $$(G, *)$$ be a group. Then $$(G, *)$$ has the Latin square property.

Proof available

THEOREM

The identity of a group is unique.

Proof available

THEOREM

$$(\mathbb{N}, +)$$ is not a group.

Proof available

THEOREM

$$(\mathbb{N}, *)$$ is not a group.

Proof available

THEOREM

$$(\mathbb{Q}, *)$$ is not a group.

Proof available

THEOREM

$$(\mathbb{R}, *)$$ is not a group.

Proof available