  We study the group structure of baryon anti-decuplet containing the
$\Theta^+$. We derive the SU(3) mass relations among the pentaquark baryons in
the anti-decuplet, when there is either no mixing or ideal mixing with the
pentaquark octet, as advocated by Jaffe and Wilczek. This constitutes the
Gell-Mann--Okubo mass formula for the pentaquark baryons. We also derive SU(3)
symmetric Lagrangian for the interactions of the baryons in the anti-decuplet
with the meson octet and the baryon octet. Our analysis for the decay widths of
the anti-decuplet states suggests that the N(1710) is ruled out as a pure
anti-decuplet state, but it may have anti-decuplet component in its
wavefunction if the multiplet is mixed with the pentaquark octet.
