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[That means every group is a supergroup but not every supergroup is a group.]
One definition of it is as the structure generated by a superalgebra. It is also a special case of a quantum group.
One of the main uses of supergroups is in the study of supersymmetry. In fact, some physicists use the word supersymmetry to describe ANY supergroup, not just the one associated with the Poincaré group, making both words practically synonymous.