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The attempts to construct a counterexample to the Strong Perfect Graph Conjecture yielded the notion of partitionable graphs as minimal imperfect graphs; then near-factorizations of finite groups gained some interest since from any near-factorization some partitionable graphs can be constructed in a natural way. Recently, the proof of SPGC was declared by Chudnovsky, Robertson, Seymour and Thomas [3], but near-factorizations remain interesting on their own rights as (i) rare objects being “close” to factorizations of groups; and (ii) they yield graphs with surprising properties.

Abstract.:  In this paper we investigate the action of a p-group G on its powerful normal subgroup N. We shall mainly be concerned to find conditions which guarantee that G acts uniserially on N. We shall also study what consequences uniserial action on N has on the structure of N.

Abstract. A subgroup H of a p-group G is n-uniserial if for each i = 1, . . . , n, there is a unique subgroup K_i such that H ≤ K_i and |K_i : H | = pⁱ. In case the subgroups of G containing H form a chain we say that H is uniserially embedded in G. We prove that if p is odd and that K is a 2-uniserial subgroup of order p in the p-group G. Then K is uniserially embedded in G. We also show that for p > 3, if K is a 2-uniserial cyclic subgroup of the p-group G, then K is uniserially embedded in G. We prove the following two theorems : (1) Let A be a soft subgroup of index greater than p. Let N₁ = N_G (A) and R = G'Z (N₁). Then the A-invariant subgroups of R containing Z (N₁) form a chain. (2) Suppose that the p-group G has a uniserially embedded subgroup P of order p. Then either G has a cyclic subgroup of index p or is of maximal class (coclass 1).