More formally, a Lie supergroup is a supermanifold G together with a multiplication morphism
This means that, formulated as commutative diagrams, the usual associativity and inversion axioms of a group continue to hold.
An equivalent algebraic approach starts from the observation that a supermanifold is determined by its ring of supercommutative smooth functions, and that a morphism of supermanifolds corresponds one to one with an algebra homomorphism between their functions in the opposite direction, i.e. that the category of supermanifolds is opposite to the category of algebras of smooth graded commutative functions.
Reversing all the arrows in the commutative diagrams that define a Lie supergroup then shows that functions over the supergroup have the structure of a Z2-graded Hopf algebra.
Likewise the representations of this Hopf algebra turn out to be Z2-graded comodules.
It can be identified with the Hopf algebra of graded differential operators at the origin.
It only gives the local properties of the symmetries i.e., it only gives information about infinitesimal supersymmetry transformations.
Using the language of schemes, which combines the geometric and algebraic point of view, algebraic supergroup schemes can be defined including super Abelian varieties.