An introduction to Simple Groups

 A group G is simple if it has only two normal subgroups, the identity element by itself and the entire group.

 

Lemma: If G has order p where p is prime, it must be simple.

 

Proof: Earlier, we proved that subgroups of finite groups have to have an order that divides the order of the group. If the order of the group is prime p, it has two divisors, 1 and p. A group or subgroup of order 1 can only be the identity, and the subgroup of order p is the improper subgroup, G itself. Not only are these the only normal subgroups, they are the only subgroups, normal or not.

 

Corollary: Any group of prime order p is isomorphic to (Zp,+).

 

This means any prime order group must be abelian. There are non-abelian simple groups, and we will see the smallest of them by the end of the month.