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group [2015/04/16 19:16]
nikolaj
group [2015/04/16 19:17] (current)
nikolaj
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 === Alternative definitions === === Alternative definitions ===
 +== Sharper definitions ==
 +We could just define left units and left inverses and prove from the group axioms that they are already units and inverses.
 +
 == Group axioms explicitly in the first order language ​ == == Group axioms explicitly in the first order language ​ ==
 Let $\langle G,* \rangle $ be a set $G$ with a binary operation. Let $\langle G,* \rangle $ be a set $G$ with a binary operation.
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 For given $G$, the set $\text{group}(G)$ is the set of all pairs $\langle G,* \rangle$, containing $G$ itself, as well a binary operation which fulfills the group axioms. One generally calls $G$ the group, i.e. the set with respect to which the operation "​$*$"​ is defined. ​ For given $G$, the set $\text{group}(G)$ is the set of all pairs $\langle G,* \rangle$, containing $G$ itself, as well a binary operation which fulfills the group axioms. One generally calls $G$ the group, i.e. the set with respect to which the operation "​$*$"​ is defined. ​
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-== Sharper definitions == 
-We could just define left units and left inverses and prove from the group axioms that they are already units and inverses. 
  
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 === Subset of === === Subset of ===
 [[Monoid]], [[Loop]] [[Monoid]], [[Loop]]
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