Every Group of Order Five or Smaller is Abelian Proof

Every Group of Order Five or Smaller is Abelian Proof. In this video we prove that if G is a group whose order is five or smaller, then G must be abelian.

[MrCoreyTexas]

Never saw this result until your video. Love learning something truly new to me!

[MrCoreyTexas]

So after watching this video, you start to think about the other groups with small non-prime orders, and naturally the question arises what about groups with order 6? Well you have the symmetric group of 3, which has 3!=3*2*1=6 elements, the 6 permuations of 1, 2, and 3. You'll see that this particular group is not abelian. It's always easy to find an abelian group, you just take Fancy Z additive modulo n which is cyclic and abelian.

[MrCoreyTexas]

So the Klein 4-group is not cyclic and has subgroups of order 2... Trying to think of a group with order 4 with every element having an order of 1? Ive come to the conclusion that no such thing exists, because every element having order 1 would be the identity element, and you can't have a group with 4 identity elements, the duplicate elements have to be removed.

[ptyamin]

why must the order of every element in the group divide the order of the group?

[drvcsrivastava]

Whether an abelian group of order 6 must be cyclic

[toaj]

Case II for |G|=4 is the Klein 4 Group right?