Question

Prove that every group which has three or four elements us abelian.

Answer 1

A group of order 3 is cyclic, since 3 is a prime. Therefore it is commutative.
Let us consider a group G of order 4. The order of every element
a divisor of o(G). The divisors of 4 bre 1, 2 and 4.
Case 1:
If there exists an element a of order 4 in G then the group is cyclic
and therefore it is commutative.
Case 2:
If there exists no element of order 4, then each non-identity element of the group is of order 2 and the order of the identity element is 1.
Therefore for every element a in G, aoa = e. Therefore a = a-1 for all a in G.
Let a€G,b€G. Then a = a-1, b = b-1. aob E G and aob=(aob)-1= b-1oa-1 = boa.
As aob = boa for all a, b in G, G is commutative. It follows that a group
of order 4 is always commutative.

Hence proved.