Lie supergroup

The concept of supergroup is a generalization of that of group. In other words, every supergroup carries a natural group structure, but there may be more than one way to structure a given group as a supergroup. A supergroup is like a Lie group in that there is a well defined notion of smooth function defined on them. However the functions may have even and odd parts. Moreover, a supergroup has a super Lie algebra which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point for classification.

More formally, a Lie supergroup is a supermanifold G together with a multiplication morphism ${\displaystyle \mu :G\times G\rightarrow G}$, an inversion morphism ${\displaystyle i:G\rightarrow G}$ and a unit morphism ${\displaystyle e:1\rightarrow G}$ which makes G a group object in the category of supermanifolds. This means that, formulated as commutative diagrams, the usual associativity and inversion axioms of a group continue to hold. Since every manifold is a super manifold, a Lie supergroup generalises the notion of a Lie group.

...
Wikipedia