Normal subgroups of multiply transitive permutation groups

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The Modular Homology of Inclusion Maps and Group Actions

Communictated by the Managing Editors Let 0 be a finite set of n elements, R a ring of characteristic p>0 and denote by Mk the R-module with k-element subsets of 0 as basis. The set inclusion map: Mk Mk&1 is the homomorphism which associates to a k-element subset 2 the sum (2)=11+12+}}}+1kof all its (k&1)-element subsets 1i. In this paper we study the chain 0 M 0 M 1 M 2}}}M k M k+1 M k+2}}} (*) arising from. We introduce the notion of p-exactness for a sequence and show that any interval of (*) not including Mn 2 or Mn+1 2 respectively, is p-exact for any prime p>0. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups on 0 of order not divisible by p. If an interval of (*) , or an equivalent sequence arising from a permutation group on 0, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application are p-rank formulae for orbit inclusion matrices. 1996 Academic Press, Inc. 1

Metric intersection problems in Cayley graphs and the Stirling recursion

In the symmetric group Sym(n) with n at least 5 let H be a conjugacy class of elements of order 2 and let \Gamma be the Cayley graph whose vertex set is the group G generated by H (so G is Sym(n) or Alt(n)) and whose edge set is determined by H. We are interested in the metric structure of this graph. In particular, for g\in G let B_{r}(g) be the metric ball in \Gamma of radius r and centre g. We show that the intersection numbers \Phi(\Gamma; r, g):=|\,B_{r}(e)\,\cap\,B_{r}(g)\,| are generalized Stirling functions in n and r. The results are motivated by the study of error graphs and related reconstruction problems.Comment: 18 page