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

Definition of a Group

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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  , then

1. Closure
 
2. Associativity
 
3. Identity
 
4. Inverse
 

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

Order of a Group

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

Diagram

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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

References

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  1. Binary operation at wikipedia