Question

Prove that a group with three elements is necessarily commutative

Answer 1

Answer:

There is only one group of order 3, the cyclic group of order 3 (which is Abelian).

Proof: Let e be the identity element, # the group operation, and g an element of the group other than e. Then g#g is not e, otherwise the order of g would be 2 but 2 does not divide the order of the group. Also g#g is not g, otherwise g would be e. So e, g, and g#g are the three distinct elements of the group. Similarly we can show that g#g#g is not g or g#g so it must be e. Thus g is a generator of the group and has order 3. (g#g is another generator.)