redundancy of two-sidedness in definition of group

Primary tabs

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).

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.

Major Section:

Reference

Type of Math Object:

Definition

Parent:

Groups audience: