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

From Wikibooks, open books for an open world
Jump to: navigation, search
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 \ast


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

Usages[edit]

  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[edit]

  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