Some Properties of Uniserially Embedding of Subgroups of p-Groups

Abstract

This paper focuses attention on the study of the Question 3.1. of [1] and it can be considered as a continuation of the previously mentioned paper. 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 H is an n-uniserial subgroup of a cyclic p-group G, then H is uniserially embedded in G. We also show that if H is an n-uniserial subgroup of the p-group G such that |G| ≤ p⁵, then H is uniserially embedded in G and we determine that if H is a 1-uniserial subgroup of order p² in the p-group G of order p⁵ and C_G(H) = H, then H is uniserially embedded in G.

Key words and phrases. n-uniserial subgroup, p-group, soft subgroup, uniserially embedding.

2000 Mathematics Subject Classification. 20D15