Consequences of the group axioms

This article contains consequences that come from the axioms of the group.

Consequences

Let - be a group with a binary operation , that satisfies the axioms of the group. Then the following consequences from the axioms of the group are valid:

1. There is only one identity element e_G in the group.
2. For any element there is only one single inverse element .
3. For any elements the equality follows the equality . This consequence means that there may be a reduction in the group. In this case, we reduced both parts of the equality by the element g.
4. For any elements an equation of the type: has only one solution: .
5. For any elements an equation of the type: has only one solution: .
6. For any elements there is an inverse element - for the product of elements: g₁, g₂.