redundancy of two-sidedness in definition of group

In the definition of group, one usually supposes that there is a two-sided identity elementMathworldPlanetmath and that any element has a two-sided inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (cf. group (

The group may also be defined without the two-sidednesses:

A group is a pair of a non-empty set G and its associative binary operationMathworldPlanetmath (x,y)xy such that

1) the operationMathworldPlanetmath has a right identity element e;

2) any element x of G has a right inverse x-1.

We have to show that the right identity e is also a left identity and that any right inverse is also a left inverse.

Let the above assumptionsPlanetmathPlanetmath on G be true.  If a-1 is the right inverse of an arbitrary element a of G, the calculation


shows that it is also the left inverse of a.  Using this result, we then can write


whence e is a left identity element, too.

Title redundancy of two-sidedness in definition of group
Canonical name RedundancyOfTwosidednessInDefinitionOfGroup
Date of creation 2015-01-20 17:28:03
Last modified on 2015-01-20 17:28:03
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 3
Author pahio (2872)
Entry type Definition