Prove every group of order less or equal to five is abelian Is it possible to prove that every group of order less or equal to five is abelian?   
We know that groups of prime order are cyclic and therefore commutative. As the number $4$ is the only composite number $\le5$, it basically remains to show this for groups of order four.

Group of order 1 is trivial, groups of order 2,3,5 are cyclic by lagrange theorem so they are abelian. 
For a group of order 4, if it has an element of order 4, it is abelian since it is cyclic(isomorphic to Z4).
If orders of every element are 2, then the inverse of an element is the element itself so you can verify every element commute with each other so the group is abelian(isomorphic to klein four group)
You can make the cayley table(operation table) for the group of order 4. You will find 4 possibilities. 3 of them are just same groups with different elements' name and are all isomorphic to Z4.
The other one is the klein four group. Those are all abelian if you check the table
