  Let G be a topological group. Then the based loopspace of G is an algebra
over the cacti operad, while the double loopspace of the classifying space of G
is an algebra over the framed little discs operad. This paper shows that these
two algebras are equivalent, in the sense that they are weakly equivalent
E-algebras, where E is an operad weakly equivalent to both framed little discs
and cacti. We recover the equivalence between cacti and framed little discs,
and Menichi's isomorphism between the BV-algebras obtained by taking the
homology of the loopspace of G and of the double loopspace of BG.
