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Abstract Algebra/Group Theory/Group/Definition of a Group
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Definition of a Group
Firstly, a Group is
a non-empty set, with a binary operation.Binary operation at wikipedia
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
Order of group G, o(G), is the number of distinct elements in G
Diagram
frame| Closure: a*b is in G if a, b are in Group G frame| Associativity: (a*b)*c = a*(b*c) if a, b, c are in Group G frame| Identity: 1. Group G has an identity eG. 2. eG*c = c*eG = c if c is in Group G frame| Inverse: 1. if c is in G, c-1 is in G. 2. c*c-1 = c-1*c = eG
References