Sylow Theorems/Examples/Sylow 2-Subgroups in Group of Order 12

Example of Use of Sylow Theorems
In a group of order $12$, there are either $1$ or $3$ Sylow $2$-subgroups.

Proof
Let $G$ be a group of order $12$.

Let $n_2$ be the number of Sylow $2$-subgroups in $G$.

From the Fourth Sylow Theorem, $n_2$ is congruent to $1$ modulo $2$, that is, odd.

Let $H$ be a Sylow $2$-subgroup of $G$.

We have that:
 * $12 = 3 \times 2^2$

and so the order of $H$ is $4$.

Thus:

From the Fifth Sylow Theorem:
 * $n_2 \divides 3$

where $\divides$ denotes divisibility.

Thus there may be $1$ or $3$ Sylow $2$-subgroups of $G$.