A characterization of ZM-groups

TL;DR

This paper characterizes ZM-groups using specific functions and proves a related conjecture, contributing to the understanding of their structure.

Contribution

It provides a new characterization of ZM-groups based on functions, and proves Conjecture 6 from prior work.

Findings

Characterization of ZM-groups established

Proof of Conjecture 6 in [4]

Enhanced understanding of ZM-group functions

Abstract

In this short note we give a characterization of ZM-groups that uses the functions defined and studied in [3,4]. This leads to a proof of Conjecture 6 in [4].

Equations

$$\phi (G)=|\{a\in G\mid o(a)=\exp (G)\}|\text{mbox}andS(G)=\sum _{H\le G}\phi (H).$$

$$n_d^{\prime }=n_d\phi (d)$$

$$S(G)\ge \sum _{H\in C(G)}\phi (H)=\sum _{H\in C(G)}\phi (|H|)=\sum _{d\mid |G|}{n}_d\phi (d)=\sum _{d\mid |G|}{n}_d^{\prime }=|G|,$$

$$\mathrm{ZM}(m,n,r)=⟨a,b\mid a^m=b^n=1,b^{-1}ab=a^r⟩,$$

$$\gcd (m,n)=\gcd (m,r-1)=1\text{mbox}and{r}^n\equiv 1(\mathrm{mod}m).$$

$$L=\left\{(m_1,n_1,s)\in {\mathbb{N}}^3\mid m_1|m,n_1|n,s<m_1,m_1|s\frac{r^n-1}{r^{n_1}-1}\right\}.$$

$$H_{(m_1,n_1,s)}=\bigcup _{k=1}^{\frac{n}{n_1}}\alpha (n_1,s{)}^k⟨a^{m_1}⟩=⟨a^{m_1},\alpha (n_1,s)⟩,$$

$$|H_{(m_1,n_1,s)}|=\exp (H_{(m_1,n_1,s)})=\frac{mn}{m_1{n}_1},$$

$$\phi (H_{(m_1,n_1,s)})\ne 0\text{mbox}ifandonlyif{H}_{(m_1,n_1,s)}\text{mbox}iscyclic.$$

$$S(G)=\sum _{H\in C(G)}\phi (H)=|G|,$$

Full text

MSC (2010): Primary 20D60, 11A25; Secondary 20D99, 11A99.

Key words: Gauss formula, Euler’s totient function, finite group, order of an element, exponent of a group.

1 Introduction

Given a finite group $G$, we consider the functions

[TABLE]

The class of finite groups $G$ for which $S(G)=|G|$ has been partially determined in [4]. We are now able to complete this study by proving the following result.

Theorem 1.1. Let $G$ be a finite group. Then $S(G)\geq|G|$, and we have equality if and only if $G$ is a ZM-group, i.e. a group with all Sylow subgroups cyclic.

In particular, since a ZM-group is nilpotent if and only if it is cyclic, we obtain the result in Conjecture 6 of [4].

Corollary 1.2. Let $G$ be a finite nilpotent group. Then $S(G)\geq|G|$, and we have equality if and only if $G$ is cyclic.

2 Proof of Theorem 1.1

Let $C(G)$ be the poset of cyclic subgroups of $G$. For every divisor $d$ of $|G|$ we denote by $n_{d}$ the number of cyclic subgroups of order $d$ of $G$ and by $n^{\prime}_{d}$ the number of elements of order $d$ in $G$. Then we have

[TABLE]

because a cyclic subgroup of order $d$ contains $\varphi(d)$ elements of order $d$. One obtains

[TABLE]

as desired.

Assume now that $S(G)=|G|$. Then $\varphi(H)=0$ for all non-cyclic subgroups $H$ of $G$. Since for any $p$-group $P$ we have $\varphi(P)\neq 0$ (see [3]), it follows that all Sylow subgroups of $G$ are cyclic, that is $G$ is a ZM-group.

Conversely, assume that $G$ is a ZM-group. By [2] such a group is of type

[TABLE]

where the triple $(m,n,r)$ satisfies the conditions

[TABLE]

It is clear that $|{\rm ZM}(m,n,r)|=mn$. The subgroups of ${\rm ZM}(m,n,r)$ have been completely described in [1]. Set

[TABLE]

Then there is a bijection between $L$ and the subgroup lattice $L({\rm ZM}(m,n,r))$ of ${\rm ZM}(m,n,r)$, namely the function that maps a triple $(m_{1},n_{1},s)\in L$ into the subgroup $H_{(m_{1},n_{1},s)}$ defined by

[TABLE]

where $\alpha(x,y)=b^{x}a^{y}$, for all $0\leq x<n$ and $0\leq y<m$. We can easily check that

[TABLE]

and so

[TABLE]

This shows that

[TABLE]

completing the proof.