We study ways of distinguishing models with Z_2-stabilized dark matter (DM) from models where the DM becomes stable by other symmetries, employing the latter to be a Z_3 symmetry for illustration. The idea is based on the observation that a Z_3-charged mother particle can decay into one or two DM particles together with Standard Model particle(s), whereas a Z_2-charged one (typically) decays into a single DM particle. As main toolkits, we employ four kinematic variables such as the invariant mass variable, the M_(T2) variable, the energy spectrum of visible particles, and the M_2 variables. We emphasize that all those observables are complementary to one another, i.e., (almost) all potential scenarios arising in Z_2 and Z_3 models can be addressed by the strategies associated with them. Results from Monte Carlo simulation are presented to show the viability of the proposed techniques.
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