Differences between phase and group delay
Note that a constant group delay is not sufficient for a band-limited signal to exhibit no dispersion. It is the phase delay that needs to be constant. If the phase is affine, i.e., if we have

$$\phi(\omega)=a+b\omega,\qquad \omega&gt;0\tag{1}$$

the group delay is constant

$$\tau_g(\omega)=-\frac{d\phi(\omega)}{d\omega}=-b\tag{2}$$

but the phase delay is not constant if $a\neq 0$:

$$\tau_p(\omega)=-\frac{\phi(\omega)}{\omega}=-\frac{a}{\omega}-b\tag{3}$$

And it is the phase delay that determines the delay of a sinusoid at the output of an LTI system. For $x(t)=\sin(\omega_0t)$, the output of an LTI system with magnitude response $M(\omega)$ and phase delay $\tau_p(\omega)$ is given by

$$y(t)=M(\omega_0)\sin(\omega_0(t-\tau_p(\omega_0)))\tag{4}$$

Only for a (strictly) linear phase (i.e., for $a=0$ in Eq. $(1)$) does the phase delay equal the group delay. In that case, the delay experienced by a sinusoid at the input of an LTI system becomes independent of frequency.