= Solution
A <dispersion relation> specifies the allowed <angular frequency> $\omega(k)$ of a <wavevector> $k$. A single harmonic wave $A\cos(kx-\omega t)$ has constant-phase surfaces moving at the <phase velocity>,
$$
\boxed{v_p=\omega/k}.
$$
For two nearby components, use the identity
$$
\cos(k_1x-\omega_1t)+\cos(k_2x-\omega_2t)
=2\cos\!\left(\frac{\Delta k\,x-\Delta\omega\,t}{2}\right)
\cos(\bar kx-\bar\omega t).
$$
The slowly varying envelope moves at $\Delta\omega/\Delta k$. Taking a narrowband limit gives the <group velocity>,
$$
\boxed{v_g=\frac{d\omega}{dk}=v_p+k\frac{dv_p}{dk}}.
$$
More generally, expand $\omega(k)$ around the central <wavevector> of a wavepacket. The linear term translates its envelope at $v_g$; the quadratic and higher terms produce <group velocity dispersion> and packet distortion. This interpretation needs a sufficiently narrow spectrum and weak enough absorption and distortion for an identifiable envelope to persist.
For light in a transparent material, $k=n(\omega)\omega/c$, so
$$
\boxed{v_p=\frac cn},\qquad
\boxed{v_g=\frac{c}{n+\omega\,dn/d\omega}
=\frac{c}{n-\lambda_0\,dn/d\lambda_0}}.
$$
The derivative is evaluated at the wavepacket's central <frequency> and $\lambda_0$ is the vacuum <wavelength>. In a nondispersive medium with constant $n$, the two velocities coincide and a packet can propagate without dispersion. In normal optical dispersion $dn/d\lambda_0<0$, the displayed model gives $v_g<v_p$.
The two velocities can differ even without material dispersion: a waveguide mode with $\omega^2=\omega_c^2+c^2k^2$ has $v_p=\omega/k>c$ and $v_g=c^2k/\omega<c$, with $v_pv_g=c^2$. The <phase velocity> is not a signal speed. Near strongly absorbing resonances an anomalous derivative can also make the <group velocity> exceed $c$ or become negative; reshaping then prevents its simple identification with information transport. Relativistic causality constrains the signal front, not the speed of an arbitrary moving <phase> or distorted intensity maximum.
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