# redundancy of two-sidedness in definition of group

In the definition of group, one usually supposes that there is a two-sided identity element and that any element has a two-sided inverse (cf. group (http://planetmath.org/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 operation $(x,y)\mapsto xy$ such that

1) the operation 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 assumptions on $G$ be true.  If $a^{-1}$ is the right inverse of an arbitrary element $a$ of $G$, the calculation

 $a^{-1}a=a^{-1}ae=a^{-1}aa^{-1}(a^{-1})^{-1}=a^{-1}e(a^{-1})^{-1}=a^{-1}(a^{-1}% )^{-1}=e$

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

 $ea=(aa^{-1})a=a(a^{-1}a)=ae=a,$

whence $e$ is a left identity element, too.

Title redundancy of two-sidedness in definition of group RedundancyOfTwosidednessInDefinitionOfGroup 2015-01-20 17:28:03 2015-01-20 17:28:03 pahio (2872) pahio (2872) 3 pahio (2872) Definition