  The Dedekind eta function $\eta(\tau)$ is defined by \[\eta(\tau)=e^{\pi
i\tau/12}\prod_{n=1}^{\infty}\left(1-e^{2\pi i
n\tau}\right),\quad\text{when}\;\text{Im}\,\tau>0.\] It plays an important role
in number theory, especially in the theory of modular forms. Its
24$^{\text{th}}$ power, $\eta(\tau)^{24}$, is a modular form of weight 12 for
the modular group $\text{PSL}\,(2,\mathbb{Z})$. Up to a constant,
$\eta(\tau)^{24}$ is equal to the modular discriminant $\Delta(\tau)$. In this
note, we give an elementary proof of the transformation formula for the
Dedekind eta function under the action of the modular group
$\text{PSL}\,(2,\mathbb{Z})$. We start by giving a proof of the transformation
formula \[\eta\left(-\frac{1}{\tau}\right)=(-i\tau)^{1/2}\eta(\tau)\]using the
Jacobi triple product identity and the Poisson summation formula. Both of these
formulas have elementary proofs. After we establish some identities for the
Dedekind sum, the transformation formula for $\eta(\tau)$ under the
transformation \[\tau\mapsto \frac{a\tau+b}{c\tau+d}\]induced by a general
element of the modular group $\text{PSL}\,(2,\mathbb{Z})$ is derived by
induction.
