Some homology groups of wreathe products
Norman Blackburn · Illinois Journal of Mathematics · 1972
NoA BcLet p be a prime and for each integer n _ 1 denote by P the Sylow p- subgroup of the symmetric group of degree pn.Thus P is a group of order p, where/c 1 -p -p-1; in particular PI is the cyclic group of order p. Pn acts as a permutation group on p symbols and if these symbols form basis of an elementary Abelian p-group A, then Am is ZP-module.The split extensien of Am by P is P+I" P+ A P.In this note the groups H (P, Z) nd H1 (P, A) will be computed.I wish to express my gratitude to L. Evens for a number of discussions which have helped me considerably ia this work. Statement of resultsFor n 1, H (P1, Z) 0 since P is cyclic.For n > 1, Pn is the wreathe product of P and P_ P P P-.The calculation of H. (P, Z) will be chieved by computing the Schur multi- plier of a wreathe product G H, where G and H are arbitrary groups and G acts as in its regular representation.To state the result let T be the tensor square of the abelian group H/H'" T H/H' (R) H/H'.Let K be the subgroup of T generated by all elements of the form h H' (R) h.Hh.H' @ hl H' (h h e H ).Let G denote set of elements of G hving the property that if x e G and x 1, then G contains either x or x -1 but not both.Let G be the set of involutions in G. Let C (G; H) denote the direct sum of GI copies of T and [GI copies of T/K.TIEOE 1. H (G H, Z) is the direct sum of H. (G, Z), H (H, Z) and C(G;H).Application of this with G P, H P_I shows that H (P, Z) is the direct sum of H. (P_I, Z) and C (P P._).As is well known, P,,_/P,_ is elementary Abelian of order p-, so in this case T is elementary Abelian of