# Abstract Algebra/Group Theory/Group/Definition of a Group/Definition of Identity

Identity:
1. Group G has an identity eG.
2. eG*c = c*eG = c if c is in Group G

Let G be a group with binary operation ${\displaystyle \ast }$

${\displaystyle \exists \;e_{G}\in G:\forall \;g\in G:e_{G}\ast g=g\ast e_{G}=g}$

## Usages

1. The identity of G, eG, is in group G.
2. Group G has an identity eG
3. If g is in G, eG ${\displaystyle \ast }$ g = g ${\displaystyle \ast }$ eG = g
4. e is the identity of group G if
e is in group G, and
e ${\displaystyle \ast }$ g = g ${\displaystyle \ast }$ e = g for every element g in G.

## Notice

1. eG always mean identity of group G throughout this section.
2. G has to be a group
3. If a is not in group G, a ${\displaystyle \ast }$ eG may not equal to a
4. If ${\displaystyle \circledast }$ is not the binary operation of G, a ${\displaystyle \circledast }$ eG may not equal to a