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Let Fn be a family of subgroups of n which is closed under taking subgroups and conjugates. Such a family has a classifying space, BFn, and we showed in an earlier paper that a compatible choice of Fn for each n gives a simplicial monoid ' nBFn, which group completes to an innite loop space. In this paper we dene a ltration of the associated spectrum whose ltration quotients, given an extra condition on the families, can be identied in terms of the classifying spaces of the families of subgroups that were chosen. This gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum. We calculate two examples. The rst is related to elementary abelianp-groups, and the second gives a new expression for the desuspension ofSpm(S0)=Spm 1(S0) as a suspension spectrum. In this paper we explore the identication of innite loop spaces that arise as group completions of certain classifying spaces with monoid structures. The proto- typical example of such an innite loop space is given by the Barratt-Priddy-Quillen theorem, which states that the group completion of the monoid ' Bn is the in- nite loop space QS0. This theorem was generalized in (L), where we replaced Bn with BFn, tom Dieck's classifying space of a family of subgroups of n ,a nd then appealed to work of Segal to show that the resulting monoid ' BFn has an innite loop space as its group completion. The spaces QS0 and K(Z; 0) occur as extreme examples of the construction, but in general it is dicult to identify the innite loop space, or the associated spectrum, that arises from a specic choice of families. In this paper we give a ltration of the associated spectrum such that the ltration quotients can be identied in terms of the classifying spaces of the families of subgroups that were chosen. This gives a way to take group theoretic data about the families of subgroups and obtain homotopy theoretic data about the spectrum associated to the resulting innite loop space. To describe our results in more detail, we require some notation. For each nonnegative integer n ,l etFn be a family of subgroups ofn ,a nd letF denote the collection of families fFngn2Z+ .L etBFn be the classifying space of Fn ,w hose construction we review in Section 2. If the families are closed under block sum of permutations, then ' BFn is a simplicial monoid, and we review in Section 2 how work of Segal applies to show that its group completion is an innite loop space.

A filtration of spectra arising from families of subgroups of symmetric groups