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We introduce a new algebraic invariant Xa(M, N) of a subfactor N C M. We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant X(M, N) introduced by Y. Kawahigashi. We also show that this group is contained in the center of Out(M, N) in many interesting examples such as quantum SU(n)k subfactors with level k (k > n+ 1), but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.

Commutativity of automorphisms of subfactors modulo inner automorphisms