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
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- 2. Associativity
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- 3. Identity
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- 4. Inverse
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From now on, eG always means identity of group G.
- 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
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Associativity: (a*b)*c = a*(b*c) if a, b, c are in Group G
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Identity: 1. Group G has an identity eG. 2. eG*c = c*eG = c if c is in Group G
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Inverse: 1. if c is in G, c-1 is in G. 2. c*c-1 = c-1*c = eG
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