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

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

Let G be a [[../|group]] with [[../../../../Binary Operations|binary operation]] $\ast$

$\mathrm{\exists }{e}_G\in G:\mathrm{\forall }g\in G:e_G\ast g=g\ast e_G=g$

Usages

1. Template:AnchorThe identity of G, e_G, is in group G.
2. Template:AnchorGroup G has an identity e_G
3. Template:AnchorIf g is in G, e_G $\ast$ g = g $\ast$ e_G = g
4. Template:Anchore 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 $⊛$ is not the binary operation of G, a $⊛$ e_G may not equal to a