The subgroup-subfactor

Abstract

In this paper, we compute the standard invariant of the 'subgroup-subfactor' P×_α|HH⊂P×_αG, where α denotes an outer action of a finite group G on a II₁ factor P, and P×_α|HH denotes the obvi- ous crossed-product obtained by restricting the action to H. We then use this description to exhibit a pair of non-isomorphic subgroups H_i, i=1, 2, of the symmetric group S₄ such that the subfactors R×_α|Hi H_i⊂P×_α G, i=1, 2 are conjugate, thereby disproving a conjecture of Thomsen-see [9]-that 'the subgroup-subfactor re- members the subgroup' (provided the subgroup contains no non-trivial normal subgroup of the ambient group).