Some properties of uniserially embedding of subgroups of p-groups

Abstract EN

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-universal if for each i = 1, ..., n, there is a unique subgroup Ki such that H ≤ Ki and |Ki : H| = pi.

In case the subgroups of G containing H form a chain we say that H is universally embedded in G.

We prove that if H is an n-universal subgroup of a cyclic p-group G, then H is universally embedded in G.

We also show that if H is an n-universal subgroup of the p-group G such that |G| ≤ p5, then H is universally 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 CG(H) = H, then H is universally embedded in G.