We investigate finite solvable permutation groups in which all normal subgroups are transitive or semiregular. The motivation comes from universal algebra: such groups are examples of collapsing transformation monoids.

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Spectral synthesis on Abelian torsion groups is proved.In this paper C denotes the set of complex numbers. If G is an Abelian group then C G denotes the locally convex topological vector space of all complex valued functions defined on G, equipped with the pointwise operations and the product topology. The dual of C G can be identified with M c (G), the space of all finitely supported complex measures on G. This space is also identified with the set of all finitely supported complex valued functions on G in the following obvious way. If the point mass concentrated at the element g is denoted by δ g , then each measure x in M c (G) has a unique representation in the form

We give sufficient conditions for a finite permutation group to contain a fixed-point-free permutation of/>-power order for a given prime p.THEOREM 2. Let p be an odd prime number and a ^ 1. Ifp + 1 < b < §(/> +1), then every transitive permutation group of degree p a b contains a fixed-point-free p-element, that is,p a -bistf v .ACKNOWLEDGEMENT. The author is grateful to Peter P. Palfy for helpful discussions.