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## Minimum number of elements for a set to become group [closed]

A group is a nonempty set G on which there is defined a binary operation (a, b) → ab satisfying the following properties. Closure: If a and b belong to G, then ab is also in G; Associativity: a(bc) = (ab)c for all a, b, c ∈ G; Identity: There is an element 1 ∈ G such that a1 = 1a = a for all a in G; Inverse: If a is in G, then there is an element a−1 in G such that aa−1 = a−1a = 1.

Does this mean , we need to have atleast three elements in a set for it to be a candidate for Group ? A set containing binary numbers ( 0 , 1 ) cannot become a group ? under any operation ?

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Please read the FAQ for a detailed explanation of why your question is off-topic in this site and a list of suggestions of other sites where it'll fit better. Good luck! – Mariano Suárez-Alvarez Jul 5 2011 at 6:44
In any case: a set with exactly one element is a group in exactly one way, and every group has at least one element. – Mariano Suárez-Alvarez Jul 5 2011 at 6:46
(@sashank: to expand Mariano's comment above: saying " $\forall a, b, c \in G$ " does not mean they are assumed to be different, nor that they even exist. If you read carefully the axioms, the only point where the existence of an element is stated, is about $1$ ). – Pietro Majer Jul 5 2011 at 7:00