Several lines of argument suggest that a large fraction of the baryons in t
he universe may be in the form of warm (T similar to 10(5)-10(7) K) gas. In
particular, loose groups of galaxies may contain substantial reservoirs of
such gas. Observations of the cosmic microwave background by COBE place on
ly weak constraints on such an intragroup medium within the Local Group. Th
e idea of a Local Group corona dates back at least 40 years (Kahn & Woltjer
). Here we show that gas at T similar to (2-3) x 10(6) K (the approximate v
irial temperature of the Local Group)-extremely difficult to observe direct
ly-can in principle radiate a large enough flux of ionizing photons to prod
uce detectable H alpha emission from embedded neutral clouds. However, addi
tional constraints on the corona-the most stringent being pulsar dispersion
measures toward the Magellanic Clouds, and the timing mass-rule out an int
ragroup medium whose ionizing flux dominates over the cosmic background or
the major Local Group galaxies. A cosmologically significant coronal gas ma
ss could remain invisible to H alpha observations. More massive galaxy grou
ps could contain extensive coronae which are important for the baryon mass
and produce a strong, local ionizing flux.