Group fairness requires that different protected groups, characterized by a given sensitive attribute, receive equal outcomes overall. Typically, the level of group fairness is measured by the statistical gap between predictions from different protected groups. In this study, we reveal an implicit property of existing group fairness measures, which provides an insight into how the group-fair models behave. Then, we develop a new group-fair constraint based on this implicit property to learn group-fair models. To do so, we first introduce a notable theoretical observation: every group-fair model has an implicitly corresponding transport map between the input spaces of each protected group. Based on this observation, we introduce a new group fairness measure termed Matched Demographic Parity (MDP), which quantifies the averaged gap between predictions of two individuals (from different protected groups) matched by a given transport map. Then, we prove that any transport map can be used in MDP to learn group-fair models, and develop a novel algorithm called Fairness Through Matching (FTM), which learns a group-fair model using MDP constraint with an user-specified transport map. We specifically propose two favorable types of transport maps for MDP, based on the optimal transport theory, and discuss their advantages. Experiments reveal that FTM successfully trains group-fair models with certain desirable properties by choosing the transport map accordingly.