Associativity, For all ⁠ a⁠, ⁠ b, ⁠ c, ⁠ in ⁠ G, one has ⁠ ( a*b )*c = a*( b*c ), Identity element, An element e in G such that e*a = a⁠ and ⁠ a*e = a, Inverse element, For each a in ⁠ G⁠, there exists an element b in G such that a*b = e and ⁠ b*a = e, where e is the identity element., Abelian group, A group, where for each a and b in G, the equation a*b=b*a is true, Non-abelian group, A group, where for each a and b in G, the equation a*b=b*a is false, Order of group, The number of elements of G, Order of an element, The smalles positive integer n such that an=e, Symmetric group, Group of bijections or permutations

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