Consequences of the group axioms

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

Consequences

Let Group notation - be a group with a binary operation Binary group 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 eG in the group.
  2. For any element Any element of the group there is only one single inverse element Inverse element .
  3. For any elements Any three elements of the group the equality Equality of two elements in a group follows the equality Equality after reduction of both parts of the element. 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 Any two elements of the group an equation of the type: A linear equation in the group (the first type) has only one solution: The solution of the equation (the first type).
  5. For any elements Any two elements of the group an equation of the type: A linear equation in the group (the second type) has only one solution: The solution of the equation (the second type).
  6. For any elements Any two elements of the group there is an inverse element Inverse element to the product of two group elements - for the product of elements: g1, g2.

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