Motivated by recent experiments with Bechgaard salts, we investigate the competition between antiferromagnetism and triplet superconductivity in quasi one-dimensional electron systems. We unify the two orders in an SO(4) symmetric framework, and demonstrate the existence of such symmetry in one-dimensional Luttinger liquids. SO(4) symmetry, which strongly constrains the phase diagram, can explain coexistence regions between antiferromagnetic, superconducting, and normal phases, as observed in (TMTSF)2PF6. We predict a sharp neutron scattering resonance in superconducting samples.A common feature of many strongly correlated electron systems is proximity of a superconducting state to some kind of magnetically ordered insulating state. Examples include organic materials [1,2], heavy fermion superconductors [3,4], and high T c cuprates [5]. Several theoretical analyses suggest that a strong repulsion between the two orders plays an important role in determining the phase diagram and low energy properties of these materials [6]. The idea of competing orders was developed into the SO(5) theory of high T c superconductivity of S.C. Zhang [7]. SO(5) symmetry has also been applied to study competition of ferromagnetism and p-wave superconductivity in Sr 2 RuO 4 [8], and antiferromagnetism and d-wave superconductivity in κ-BEDT-TTF salts [9].In this paper we consider the interplay of antiferromagnetism (AF) and triplet superconductivity (TSC) in quasi one-dimensional (Q1D) electron systems. Our study is motivated by Q1D Bechgaard salts (TMTSF) 2 X. The most well studied material from this family, (TMTSF) 2 PF 6 , is an antiferromagnetic insulator at ambient pressure and a superconductor at high pressures [10,11,12,13]. The symmetry of the superconducting order parameter in (TMTSF) 2 PF 6 is not yet fully established, but there is strong evidence that electron pairing is spin triplet: the superconducting T c is strongly suppressed by disorder [14]; critical magnetic field H c2 in the interchain direction exceeds the paramagnetic limit [15]; the electron spin susceptibility, obtained from Knight shift measurements, does not decrease below T c [16]. In another material from this family, (TMTSF) 2 ClO 4 , superconductivity is stable at ambient pressure and also shows signatures of triplet pairing [17,18,19,20]. Insulator to superconductor transition as a function of pressure has also been found in(TMTSF) 2 AsF 6 [21].The phase diagram of interacting electrons in one dimension was obtained in Ref. 22 using bosonization and renormalization group (RG) analyses. At incommensurate filling, this system has a phase boundary between spin density wave (SDW) and TSC phases when K ρ = 1 and g 1 > 0 (K ρ is the Luttinger parameter in the charge sector, g 1 the backward scattering amplitude). The starting point of our discussion is the observation that, in the absence of umklapp, 1D Luttinger liquids have an "isospin" SO(4) iso symmetry [23] at the boundary between SDW and TSC phases. To define this symmetry, we introduce the charge of le...