Let H be a closed subgroup of a regular abelian paratopological group G. The group reflexion G ( of G is the group G endowed with the strongest group topology, weaker that the original topology of G. We show that the quotient G/H is Hausdorff (and regular) if H is closed (and locally compact) in G ( . On the other hand, we construct an example of a regular abelian paratopological group G containing a closed discrete subgroup H such that the quotient G/H is Hausdorff but not regular. In this paper we study the properties of the quotients of paratopological groups by their normal subgroups. By a paratopological group G we understand a group G endowed with a topology � making the group operation continuous, see (ST). If, in addition, the operation of taking inverse is continuous, then the paratopological group (G;�) is a topological group. A standard example of a paratopological group failing to be a topological group is the Sorgefrey line L, that is the real line R endowed with the Sorgefrey topology (generated by the base consisting of half-intervals (a;b), a < b). Let (G;�) be a paratopological group and HG be a closed normal subgroup of G. Then the quotient group G=H endowed with the quotient topology is a paratopological group, see (Ra). Like in the case of topological groups, the quotient homomorphism � : G ! G=H is open. If the subgroup HG is compact, then the quotient G=H is Hausdorff (and regular) provided so is the group G, see (Ra). The compactness of H in this result cannot be replaced by the local compactness as the following simple example shows. Example 1. The subgroup H = f(−x;x) : x 2 Qg is closed and discrete in the square G = L 2 of the Sorgenfrey line L. Nonetheless, the quotient group G=H fails to be Hausdorff: for any irrational x the coset (−x;x) + H cannot be separated from zero (0;0) + H. A necessary and sufficient condition for the quotientG=H to be Hausdorff is the closedness of H in the topology of group reflexion G ( of G. By the group reflexion G ( = (G;� ( ) of a paratopological group (G;�) we under- stand the group G endowed with the strongest topology � ( � � turning G into a topological group. This topology admits a categorial description: � ( is a unique topology on G such that � (G;� ( ) is a topological group; � the identity homomorphism id : (G;�) ! (G;� ( ) is continuous;