Element orders in extraspecial groups

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 Tóth and Tărnăuceanu: “For every \(a\in [0, 1]\), does there exist a sequence \((G_n)_{n\geq 1}\) of finite groups such that \( \lim_{n\to\infty} \text{cdeg} (G_n)=a\)?”. We show that such sequences are formed of finite direct products of extraspecial groups of a specific type.