  We constrain the distributions of projected radial alignment angles of
satellite galaxy shapes within the Galaxy And Mass Assembly survey group
catalogue. We identify the galaxy groups using spectroscopic redshifts and
measure galaxy projected ellipticities from Sloan Digital Sky Survey imaging.
With a sample of 3,850 groups with 13,655 satellite galaxies with high quality
shape measurements, we find a less than 2-sigma signal of radial alignments in
the mean projected ellipticity components and the projected position angle when
using galaxy shape estimates optimized for weak lensing measurements. Our
radial alignment measurement increases to greater than 3-sigma significance
relative to the expectation for no alignments if we use 2-D S\'ersic model fits
to define galaxy orientations. Our weak measurement of radial alignments is in
conflict with predictions from dark matter N-body simulations, which we
interpret as evidence for large mis-alignments of baryons and dark matter in
group and cluster satellites. Within our uncertainties, that are dominated by
our small sample size, we find only weak and marginally significant trends of
the radial alignment angle distributions on projected distance from the group
centre, host halo mass, and redshift that could be consistent with a tidal
torquing mechanism for radial alignments. Using our lensing optimized shape
estimators, we estimate that intrinsic alignments of galaxy group members may
contribute a systematic error to the mean differential projected surface mass
density of groups inferred from weak lensing observations by -1 +/- 20% at
scales around 300 kpc/h from the group centre assuming a photometric redshift
r.m.s. error of 10%, and given our group sample with median redshift of 0.17
and median virial masses ~10^{13} h^{-1}M_{sun}.
