Classification of Groups of Order up to 15
Theorem
Up to isomorphism, every group of order $\order G \le 15$ is one of the below:
| Order | Abelian | Non-Abelian |
|---|---|---|
| 1 | $\Z_1$ | |
| 2 | $\Z_2$ | |
| 3 | $\Z_3$ | |
| 4 | $\Z_4, \Z_2 \oplus \Z_2$ | |
| 5 | $\Z_5$ | |
| 6 | $\Z_6$ | $D_3 \cong S_3$ |
| 7 | $\Z_7$ | |
| 8 | $\Z_8, \Z_4 \oplus \Z_2, \Z_2 \oplus \Z_2 \oplus \Z_2$ | $D_4, \Dic 2$ |
| 9 | $\Z_9, \Z_3 \oplus \Z_3$ | |
| 10 | $\Z_{10}$ | $D_5$ |
| 11 | $\Z_{11}$ | |
| 12 | $\Z_{12}, \Z_6 \oplus \Z_2$ | $D_6, A_4, \Dic 3$ |
| 13 | $\Z_{13}$ | |
| 14 | $\Z_{14}$ | $D_7$ |
| 15 | $\Z_{15}$ |
where:
- $D_n$ is the dihedral group of order $2 n$
- $S_n$ is the $n$th symmetric group
- $A_n$ is the alternating group on $n$ points
- $\Dic n$ is the dicyclic group of order $4 n$.
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 is Cyclic Group - determines theorem for order $1$
- $(2): \quad$ Prime Group is 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 $9$
- $(4): \quad$ Group of Order p q is Cyclic - 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 $12$
$\blacksquare$
Sources
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $17$: Groups of order $15$ or less