Classification of Groups of Order up to 15

Theorem
Up to isomorphism, every group of order $$|G|\leq15 \ $$ is one of the below:

where $$D_n \ $$ is the dihedral group of order $$2n \ $$, $$A_n \ $$ is the alternating group on $$n \ $$ points, and $$Q_n \ $$ is the dicyclic group on $$2n \ $$.

Proof
The Abelian cases are the direct result of the Fundamental Theorem of Finite Abelian Groups.

The non-Abelian cases follow from seven separate theorems:


 * 1) Trivial Group - determines theorem for order 1
 * 2) Group of Prime Order Cyclic - determines theorem for orders 2, 3, 5, 7, 11, and 13
 * 3) Group of Order Prime Squared is Abelian - determines theorem for orders 4 and 9
 * 4) Cyclic Groups of Order pq - determines theorem for order 15
 * 5) Groups of Order Twice a Prime - determines theorem for orders 6, 10, 14
 * 6) Groups of Order 8 - determines theorem for order 8
 * 7) Groups of Order 12 - determines theorem for order 12