  The radio emission of a galaxy consists of thermal bremsstrahlung,
synchrotron emission from discrete supernova remnants, and diffuse synchrotron
emission from cosmic ray electrons spread over the galactic disk and halo. Each
of these components has a different spectral index so that the total radio
spectral index of a galaxy depends sensitively on their relative contribution
and on the processes shaping the diffuse synchrotron emission. In the present
paper we calculate the contribution of supernova remnants to the total
synchrotron emission and conclude that it is about 10%. This moderate
contribution has a noticeable effect on the nonthermal spectral index, lowering
it by $\simeq 0.1$ for steep spectra. We then calculate the diffuse synchrotron
emission in two simple models, a diffusion and a convection model. We find that
the spatially integrated nonthermal spectral index is in general a poor
diagnostic for the type of propagation or the importance of energy losses.
Spatially resolved radio data for the halo of galaxies are necessary in order
to draw firm conclusions. The steepening of the spectrum away from the disk is
a clear indication that synchrotron and inverse Compton losses are taking place
during the propagation of cosmic ray electrons in the halo. In those galaxies
for which spatially resolved data for the halos exist such a steepening has
been found. We conclude therefore that energy losses are generally important
and that cosmic ray electrons cannot freely escape from galaxies.
