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
![{\displaystyle \forall \;a,b\in G:a\ast b\in G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c336c4434e60a08e54ad1676c9dfc109d359e2e9)
- 2. Associativity
![{\displaystyle \forall \;a,b,c\in G:(a\ast b)\ast c=a\ast (b\ast c)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc279581a6802147bc4bc98e3bdd7af259c89d7d)
- 3. Identity
![{\displaystyle \exists \;e_{G}\in G:\forall \;g\in G:e_{G}\ast g=g\ast e_{G}=g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2678c5601a938b2fda8b92a1b9a9a716b10f69f)
- 4. Inverse
![{\displaystyle \forall \;g\in G:\exists \;g^{-1}\in G:g\ast g^{-1}=g^{-1}\ast g=e_{G}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7399cda9b496b5206f1550e547d914ce9341ac2)
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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- ↑ Binary operation at wikipedia