Definition of a Group

Firstly, a Group is

a non-empty set, with a binary operation.^[1]

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

\forall \;a,b\in G:a\ast b\in G

\forall \;a,b,c\in G:(a\ast b)\ast c=a\ast (b\ast c)

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

\forall \;g\in G:\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

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 e_G.
2. e_G*c = c*e_G = 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 = e_G