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

Let G be a group with binary operation $\ast$

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

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

Usages

1. The identity of G, e_G, is in group G.
2. Group G has an identity e_G
3. If g is in G, e_G $\ast$ g = g $\ast$ e_G = g
4. e is the identity of group G if

   e is in group G, and

   e $\ast$ g = g $\ast$ e = g for every element g in G.

Notice

1. e_G 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 $\ast$ e_G may not equal to a
4. If $\circledast$ is not the binary operation of G, a $\circledast$ e_G may not equal to a