An affirmative answer to the Jainkin-Zapirain's question

TL;DR

This paper confirms that for finite p-groups with a normal abelian subgroup, the average order of the entire group is at least as large as that of the subgroup, answering a question posed by Zapirain.

Contribution

The paper proves an improved version of Zapirain's question, establishing a lower bound for the average order in finite p-groups with normal abelian subgroups.

Findings

Confirmed that o(G) ≥ o(N) for finite p-groups with normal abelian N

Provided a positive answer to Zapirain's question in this context

Enhanced understanding of the relationship between subgroup and group average orders

Abstract

If X is a non-empty subset of a finite group G, we denote by o(x) the order of x in G. Then we put The number o(X) is called the average order of X. Zapirain in 2011 , posed the following question: Let G be a finite (p-) group and N a normal (abelian) subgroup of G. Is it true that o(G) ? Here, we will improve his question and confirm it.