Element orders in extraspecial groups

TL;DR

This paper explicitly determines element orders in extraspecial groups, explores their subgroup structures, and proves the density of cyclicity degrees in [0,1], answering a question about possible limits of this ratio.

Contribution

It provides explicit formulas for element counts in extraspecial groups and shows the density of cyclicity degrees, addressing a question about subgroup ratios in finite groups.

Findings

Number of elements of specific orders in extraspecial groups determined

Cyclicity degrees of all finite groups are dense in [0, 1]

Sequences of finite groups with prescribed cyclicity degree limits constructed

Abstract

By using the structure and some properties of extraspecial and generalized/almost extraspecial $p$-groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic subgroups of any (generalized/almost) extraspecial group. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is called the cyclicity degree of $G$ and is denoted by $cdeg(G)$. We show that the set containing the cyclicity degrees of all finite groups is dense in $[0, 1]$. This is equivalent to giving an affirmative answer to the following question posed by T\'{o}th and T\u{a}rn\u{a}uceanu: ``For every $a\in [0, 1]$, does there exist a sequence $(G_n)_{n\geq 1}$ of finite groups such that $\displaystyle\lim_{n\to\infty} cdeg(G_n)=a$?". We show that such sequences are formed of finite direct products of extraspecial groups of a specific type.