  We investigate the competition between antiferromagnetism and triplet
superconductivity in quasi one-dimensional electron systems. We show that the
two order parameters can be unified using a SO(4) symmetry and demonstrate the
existence of such symmetry in one dimensional Luttinger liquids of interacting
electrons. We argue that approximate SO(4) symmetry remains valid even when
interchain hopping is strong enough to turn the system into a strongly
anisotropic Fermi liquid. For unitary triplet superconductors SO(4) symmetry
requires a first order transition between antiferromagnetic and superconducting
phases. Analysis of thermal fluctuations shows that the transition between the
normal and the superconducting phases is weakly first order, and the normal to
antiferromagnet phase boundary has a tricritical point, with the transition
being first order in the vicinity of the superconducting phase. We propose that
this phase diagram explains coexistence regions between the superconducting and
the antiferromagnetic phases, and between the antiferromagnetic and the normal
phases observed in (TMTSF)$_2$PF$_6$. For non-unitary triplet superconductors
the SO(4) symmetry predicts the existence of a mixed phase of
antiferromagnetism and superconductivity. We discuss experimental tests of the
SO(4) symmetry in neutron scattering and tunneling experiments.
