On reflexive group topologies on abelian groups of finite exponent

The present paper deals with the existence of nondiscrete reflexive topologies on abelian groups of finite exponent, which turns out to be linked with the cardinality of the corresponding group. We prove that if the starting group G has cardinality , a reflexive topology on G must be discrete. On the other hand, if G has cardinality greater or equal than the continuum it even admits a locally compact Hausdorff group topology. We leave open the question for groups with cardinality between and .