The finite images of finitely generated groups

Abstract

Given any sequence of non-abelian finite simple primitive permutation groups Sn, we construct a finitely generated group G whose profinite completion is the infinite permutational wreath product … Sn ≀ Sn−1 ≀ … ≀ S0. It follows that the upper composition factors of G are exactly the groups Sn. By suitably choosing the sequence Sn we can arrange that G has any one of a continuous range of slow, non-polynomial subgroup growth types. We also construct a 61-generator perfect group that has every non-abelian finite simple group as a quotient. 2000 Mathematics Subject Classification: 20E07, 20E08, 20E18, 20E32.