Some further properties of soft subgroups

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 | = p ⁱ. 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 N ₁ = N _G (A) and R = G'Z (N ₁). Then the A-invariant subgroups of R containing Z (N ₁) 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).