In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is identity on the subgroup. In symbols, H is a retract of G if and only if there is an endomorphism \sigma:G \to G such that \sigma(h) = h for all h \in H and \sigma(g) \in H for all g \in G.