We know that for an ideal gas
$$PV=nRT,$$ where $P$ is pressure, $V$ volume, $n$ amount of substance in moles, and $T$ the temperature of the gas.
We can easily derive that, for a (non-free) adiabatic expansion, the product $PV^γ=\text{const.} \implies TV^{γ-1}=\text{const.}$
From the first expression we see that we can vary $V$ and $P$ such that $T$ remains constant. But from $TV^{γ-1}=\text{const.} \implies T=\frac{\text{const.}}{V^{γ-1}}$, since $V$ has varied, the temperature should not remain constant. What am I missing?
Therefore and isothermic process cannot be an adiabatic one. Is this true?