The recently discovered z=10 galaxy (Pello et al 2004) has a strong Lyman alpha emission line that is consistent with being surprisingly symmetric, even given the relatively poor quality of its spectrum. The blue wing of a Lyman alpha line originating at high redshift should be strongly suppressed by resonant hydrogen absorption along the line of sight, an expectation borne out by the observed asymmetric shapes of the existing sample of Lyman alpha emitting sources at lower redshifts (3< z < 6.7). Absorption on the blue side of the line of the Pello et al. source could be reduced if the intergalactic medium (IGM) in the vicinity of the galaxy is highly ionized, but we show that this requires an unrealistically high ionizing emissivity. We suggest instead that the Lyman alpha emitting gas be receding relative to the surrounding gas with a velocity of >= 35km/s, a large velocity that is plausible only if the galaxy is part of a larger system (group of galaxies) with a velocity dispersion >= 35km/s. We find that with this velocity shift, the observed strength and shape of the line is still consistent with the galaxy being surrounded by its own Stromgren sphere embedded in a fully neutral IGM. More generally, we predict that at any given redshift, the bright Lyman alpha emitters with broader lines would exhibit stronger asymmetry than fainter ones. Bright galaxies with symmetric Lyman alpha lines may be signposts for groups and clusters of galaxies, within which they can acquire random velocities comparable to or larger than their linewidths.