The definition of a group. Let G be a non-empty set and let ⋆ be a binary operation on G:
(bop) ⋆: G × G → G, (a, b) → a ⋆ b.
Then (G; ⋆) is a group if the following axioms are satisfied:
(G1) associativity: a ⋆ (b ⋆ c) = (a ⋆ b) ⋆ c for all a, b, c ∈ G
(G2) identity element: there exists e ∈ G such that a ⋆ e = e ⋆ a = a for all a ∈ G.
(G3) inverses: for any a ∈ G there exists a −1 (a inverse) ∈ G such that a ⋆ a−1 = a −1 ⋆ a = e.
If in addition the following holds
(G4) commutativity: a ⋆ b = b ⋆ a for all a, b ∈ G then (G; ⋆) is called an abelian group, or simply a commutative group.
Remarks: Note that (bop) is an essential part of the definition. and that (G2) must precede (G3) because (G3) refers back to the element e.
Fact: if (G; ⋆) is a group then the identity e is unique and the inverse of any a in G is uniquely determined by a.
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