I'm trying to figure out the right definition of a Lie supergroup. I'd like a definition such that (1) the tangent space at the identity is a Lie superalgebra (in the usual sense, i.e. the Lie bracket and the axioms carry the Koszul sign rule), and (2) it makes sense to ask what the parity of an element of the Lie supergroup is.

(I'm trying to understand the idea of supersymmetry as the existence of odd-parity symmetries, which should then be part of a super Lie group)

Unfortunately, I think these are incompatible: the parity of a group element ought to be locally constant, and the identity is even, so unless we add additional structure, then in a "Lie group where we also know parities", the Lie algebra doesn't get a Z/2-grading: neighborhoods of the identity have purely even grading.

So does one add additional structure to a Lie supergroup, or is asking for the degree of a non-infinitesimal symmetry not well-defined, or something else entirely?

@ArunDebray A supermanifold is more than just its points, right? It's about the set of smooth functions on the manifold. There's no "parity function" on the underlying set of a supermanifold, so I don't think there should be one for a Lie supergroup.

@MikeMiller ok, thanks! I guess it's a little weird is that one can't ask for the parity of a symmetry (does it send even elements to even elements?) -- but I guess most symmetries might not be purely even or odd, so that seems OK

@ArunDebray I guess I would go back to R^{p|q} --- what are the symmetries of that? can they be made explicit? And are any of them "signed" the way you would like?

This being the space $\Bbb R^p$ with smooth functions $C^\infty(\Bbb R^p) \otimes \Lambda(y_1, \cdots, y_q)$.