Group Definition

The group consists of $G$ , a set of elements and $\ast$, an operation that operates on that set of elements: $<G,\ast >$

The group axioms:

1. The group is closed under the operation:

$$\forall x\forall y[x\in G\wedge y\in G⟹(x\ast y)\in G]$$

2. The operation is associative for each element in the set:

$$\forall x\forall y\forall y[x\in G\wedge y\in G\wedge z\in G⟹(x\ast y)\ast z=x\ast (y\ast z)]$$

3. There exists an identity element, so that when an identity element is paired with any element via the operation, it returns that element – i.e. operation with the identity element have no effect on the element.

$$\exists e[e\in G\wedge \forall x(x\in G⟹e\ast x=x\ast e=x)]$$

4. For every element there exists an invers, where the operation between the element and its inverse result in the identity element:

$$\forall x[x\in G⟹\exists x^{-1}(x^{-1}\in G\wedge x\ast x^{-1}=x^{-1}\ast x=e)]$$