Classification of Groups of Order up to 15

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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