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 3850 groups with 13 655 satellite galaxies with high quality shape measurements, we find a less than 2σ 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 3σ significance relative to the expectation for no alignments if we use 2D Śersic 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 misalignments 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 per cent at scales around 300 h-1 kpc from the group centre assuming a photometric redshift rms error of 10 per cent, and given our group sample with median redshift of 0.17 and median virial masses ~1013 h-1Ṁ © 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.