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

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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


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a*b is in G if a, b are in Group G
(a*b)*c = a*(b*c)
if a, b, c are in Group G
1. Group G has an identity eG.
2. eG*c = c*eG = c if c is in Group G
1. if c is in G, c-1 is in G.
2. c*c-1 = c-1*c = eG


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  1. Binary operation at wikipedia