Classification of Groups of Order up to 15

Theorem
Up to isomorphism, every group of order $|G| \le 15$ is one of the below:

where:
 * $D_n$ is the dihedral group of order $2n$;
 * $A_n$ is the alternating group on $n$ points;
 * $Q_n$ 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:


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