If $X_1, \ldots, X_n \sim N(\mu, \sigma^2)$, then $ \frac{n - 1}{\sigma^2}S^2 \sim \chi^2_{n - 1} $ where $S^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i^2- \bar{x})^2$, and there's a direct relationship between the $\chi^2_p$ and Gamma($\alpha, \beta$) distributions: $ \chi^2_{n - 1} = \text{Gamma}(\tfrac{n-1}{2}, 2). $ But then why is $ S^2 \sim \text{Gamma}(\tfrac{n-1}{2}, \tfrac{2\sigma^2}{n-1}) \,? $ And why do we multiply the reciprocal with $\beta$ and not $\alpha$? Is it because $\beta$ is the scale parameter?
In general, are there methods for algebraic manipulation around the "$\sim$" other than the standard transformation procedures? Something that uses the properties of location/scale families perhaps?