Abstract algebra
Group theory

A group is a set G {\displaystyle G} together with a "group product", a binary operation ⋅: G × G → G {\displaystyle \cdot:G\times G\rightarrow G}. The group satisfies the following defining axioms (c.f. Group (mathematics) § Definition):

Identity: there exists an element e {\displaystyle e} such that, for each element a {\displaystyle a} in G {\displaystyle G}, it holds that e ⋅ a = a ⋅ e = a {\displaystyle e\cdot a=a\cdot e=a}.

Inverse: for each element a {\displaystyle a} of G {\displaystyle G}, there exists an element b {\displaystyle b} so that a ⋅ b = b ⋅ a = e {\displaystyle a\cdot b=b\cdot a=e}.

Associativity: for each triplet of elements a, b, c {\displaystyle a,b,c} in G {\displaystyle G}, it holds that (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}.
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