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

Definition of a Group

Firstly, a Group is

a non-empty set, with a binary operation.

Secondly, if G is a Group, and the binary operation of Group G is $\ast$, then

1. Closure

$\mathrm{\forall }a,b\in G:a\ast b\in G$

2. Associativity

$\mathrm{\forall }a,b,c\in G:(a\ast b)\ast c=a\ast (b\ast c)$

3. Identity

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

4. Inverse

$\mathrm{\forall }g\in G:\mathrm{\exists }{g}^{-1}\in G:g\ast g^{-1}=g^{-1}\ast g=e_G$

From now on, e_G always means identity of group G.

Order of a Group

Order of group G, o(G), is the number of distinct elements in G

Diagram

Closure: a*b is in G if a, b are in Group G

Associativity: (a*b)*c = a*(b*c) if a, b, c are in Group G

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

Inverse: 1. if c is in G, c-1 is in G. 2. c*c-1 = c-1*c = eG