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



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


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

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 $$\ast$$ g = g $$\ast$$ eG = 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) 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 $$ \ast $$ eG may not equal to a
 * 4) If $$\circledast$$ is not the binary operation of G, a $$\circledast$$ eG may not equal to a