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group

 Definition/Summary A group is a set S with a binary operation S*S -> S that is associative, that has an identity element, and that has an inverse for every element, thus making it a monoid with inverses, or a semigroup with an identity and inverses. The number of elements of a group is called its order, and the minimum power of an element that will yield the identity is that element's order. The identity's order is 1, and every other group element's order is greater.

 Equations Associativity: $\forall a,b,c \in S ,\ (a \cdot b) \cdot c = a \cdot (b \cdot c)$ Identity e: $\forall a \in S,\ e \cdot a = a \cdot e = a$ Inverse: $\forall a \in S,\ \exists a^{-1} \in S,\ a \cdot a^{-1} = a^{-1} \cdot a = e$

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