$\mathcal{Q}$-groups satisfy the equation $T_G(r,s)=0$

TL;DR

This paper proves that certain $ ext{Q}$-groups, which satisfy a specific equation involving element numbers, are solvable groups, expanding understanding of their algebraic structure.

Contribution

It establishes that $ ext{Q}$-groups satisfying $T_G(r,s)=0$ are solvable, clarifying their algebraic properties and broadening previous classifications.

Findings

$ ext{Q}$-groups satisfying $T_G(r,s)=0$ are solvable.

The result applies to groups not necessarily nilpotent.

Provides insight into the structure of groups defined by element number equations.

Abstract

The present note shows that $\mathcal{Q}$-groups in [H. Heineken and F.G. Russo, Groups described by element numbers, Forum Math. 27 (2015), 1961--1977] are solvable groups (not necessarily nilpotent) for which the equation $T_G(r,s)=0$ is satisfied.

Equations

$$L_m(G)=\{x\in G|x^m=1\}$$

$$c_k=|\{C\le G|C\text{mbox}isacyclicsubgroupoforderk\}|$$

$$|L_m(G)|=\sum _{k|m}{c}_k\phi (k)=m\cdot f(m),$$

$$T_G(r,s)=\sum _{k|n}{g}_{k,n/k}^{r,s}k(f(k)-1),$$

$${\mathbf{b}}_m(G)=|L_m(G)|\cdot m^{-1}$$

$$f(m)={\mathbf{b}}_m(G)$$

$$f(k)\leq{\big{(}g_{k,\ n/k}^{r,s}\big{)}}^{-\frac{1}{2}}\ \ \Rightarrow\ \ g_{k,n/k}^{r,s}\ f(k)\leq\sqrt{g_{k,n/k}^{r,s}}$$

$$\sum _{k|n}{g}_{k,n/k}^{r,s}f(k)k\le \sum _{k|n}\sqrt{g_{k,n/k}^{r,s}}k\le \sum _{k|n}{g}_{k,n/k}^{r,s}k$$

$$\sum _{k|n}{g}_{k,n/k}^{r,s}k(f(k)-1)\le 0$$

Full text

1. Two different approaches of investigation for the local breadth

The present note deals with finite groups only. It is shown that the $\mathcal{Q}$-groups in [7, Page 1962] and the groups of Meng and Shi in [1, 10, 11] satisfy the equation $T_{G}(r,s)=0$, mentioned in [5, Questions 1.4 and 1.5].

Chronologically [5] appears two years after [7], but even [2, 6, 12, 13] do not mention the so called “Inverse Problem to Frobenius’ Theorem” in the study of structural properties of groups by restrictions of numerical nature. A brief historical remark is then appropriate. Frobenius [3, 4] and Yamaki [9, 14, 15, 16, 17] worked on the following problem long time ago:

Problem 1.1**.**

Given a group $G$ and an integer $m\geq 1$ dividing the exponent $\exp(G)$, define

[TABLE]

and study relations between $m$, $|L_{m}(G)|$ and the structure of $G$.

Frobenius showed that a divisor $m$ of $|G|$ divides also $|L_{m}(G)|$, that is, $|L_{m}(G)|=mk$ for some $k\geq 1$, on the other hand, he conjectured that for $k=1$, then the $m$ elements of $L_{m}(G)$ form a characteristic subgroup of $G$. This was indeed shown by Iiyori and Yamaki in their aforementioned contributions with the help of the classification of simple groups. The elegance of the proof of Frobenius’ Conjecture cannot be summarized here, but its importance is clear since no restriction on $G$ is given (of any nature !). A more recent problem, studied in [1, 7, 8, 10, 11], is to look at the size of $k$, and look for structural conditions on $G$. More precisely:

Problem 1.2** (Inverse Problem to Frobenius’ Theorem).**

Classify all groups $G$ such that $m^{-1}\ |L_{m}(G)|\leq\psi(m)$ for all $m$ dividing $\exp(G)$, where $\psi:m\in\mathbb{N}\mapsto\psi(m)\in\mathbb{N}$ is a prescribed function depending only on $m$.

Now we change completely perspective and look numerically at the size of $L_{m}(G)$. Consider the Euler function $\varphi(k)$ and define the number

[TABLE]

of cyclic subgroups of $G$ of order exactly $k$. One can see that

[TABLE]

where $f(m)\geq 1$ is an integer depending only on $m$ and $G$.

Denoting by $r,s$ two real numbers and by $n=|G|$, Garonzi and Patassini [5, Page 683, §2.4] worked on the function

[TABLE]

where $g_{k,\ n/k}^{r,s}$ are the coefficient expressed by the expansion [5, (1) of Lemma 7]. These coefficients turn out to be non-negative and the case $g_{k,\ n/k}^{r,s}=0$ is completely described by [5, final part of (1) in Lemma 7].

Restrictions on $T_{G}(r,s)$ allows us to detect nilpotency in $G$ by [5, Theorem 5] and further properties were also noted by De Medts and Tǎrnǎuceanu in [2, 12, 13]. In particular, [5, Questions 1.4] asks whether the equality $T_{G}(r,s)=0$ detects solvability for $G$ for some $(r,s)$, while [5, Questions 1.5] asks whether the equality $T_{G}(r,s)=0$ implies structural properties for $G$ for some $(r,s)\in\mathbb{R}^{2}$.

2. The main result and its proof

In order to attack Problem 1.2, the authors of [1, 7, 8, 10, 11] studied what is called the local breadth, that is,

[TABLE]

of $G$ (under the general condition that $m$ is a divisor of the exponent of $|G|$, so not necessarily of $|G|$). In particular, we get

[TABLE]

comparing (1.1) with (1.2).

A first body of results is the following, where only nilpotent groups appear.

Theorem 2.1** (See [10], Main Theorem and [8], list in §4).**

Let $G$ be a group and $m\geq 1$ a divisor of $\exp(G)$. Then $f(m)\leq 2$ if and only if

• (i)

$G$* is cyclic;*

• (ii)

$G\simeq C_{h}\times C_{2^{k-1}}\times C_{2}$* with $h$ odd and $k\geq 2$;*

• (iii)

$G\simeq C_{h}\times Q_{8}=C_{h}\times\langle a,b\ |\ a^{4}=1,a^{2}=b^{2},b^{-1}ab=a^{-1}\rangle$* with $h$ odd;*

• (iv)

$G\simeq C_{h}\times\langle a,b\ |\ a^{2^{t-1}}=b^{2}=1,b^{-1}ab=a^{1+2^{t-2}}\rangle$* with $t\geq 4$ and $h$ odd;*

• (v)

$G\simeq C_{h}\times\langle a,b\ |\ a^{3}=b^{2^{s}}=1,b^{-1}ab=a^{-1}\rangle$* with $s\geq 1$ and $\gcd(h,6)=1$.*

One can see without difficulties that the previous result remains true when $m$ is a divisor of $|G|$. After Theorem 2.1, Chen, Meng and Shi [1, Theorems 1.1, 1.2] improved the classification based on the bound $f(m)\leq 2$ with another one, based on the bound $f(m)\leq 3$. Successively it was introduced the more comprehensive notion of $\mathcal{Q}$-$group$ in [7] (i.e.: a group $G$ such that $f(m)\leq m$ with $m$ divisor of $\exp(G)$) and corresponding classifications are presented in [7, Theorems 3.2, 3.5, 3.6, 3.8, 3.12, 3.14], which we are going to summarize below.

Theorem 2.2** (Structure of $\mathcal{Q}$-groups, see [7]).**

Let $G$ be a group and $m\geq 1$ a divisor of $\exp(G)$ such that $f(m)\leq m$.

• (i)

$G$* is solvable;*

• (ii)

$G$* has a $2$-nilpotent normal subgroup $M$ of index $|G:M|\in\{1,3\}$;*

• (iii)

if $p>3$ is a prime and $G$ a $p$-group, then $G$ is the product of two cyclic groups with trivial intersection;

• (iv)

if $\gcd(|G|,6)=1$, then $G$ is metabelian and the quotient $G/F(G)$ through the Fitting subgroup $F(G)$ is cyclic;

• (v)

$G=\mathrm{SL}(2,3)$* is a solvable non-metabelian $\mathcal{Q}$-group with $\gcd(|G|,6)\neq 1$.*

The notation for the Fitting subgroup, the notion of $p$-nilpotent group and the notation for the special linear group of dimension $2$ over the field with $3$ elements are very common and recalled in most of the references in the bibliography. Thanks to Theorem 2.2, we are in the position to show that a large class of solvable groups, not necessarily nilpotent, satisfies the equation $T_{G}(r,s)=0$.

Theorem 2.3**.**

If $(r,s)\in\mathbb{R}^{2}$ satisfy $1\leq g_{k,\ n/k}^{r,s}$ and f(k)\leq{\big{(}g_{k,\ n/k}^{r,s}\big{)}}^{-\frac{1}{2}}\leq k for all $k$ divisors of $n$, then $T_{G}(r,s)=0$ and $G$ is solvable.

Proof of Theorem 2.3.

It is useful to note that (1.3) becomes equal to zero when $f(k)=1$ and $f(k)=1$ happens if and only if $G$ satisfies (i) of Theorem 2.1, that is, $G$ is cyclic. Therefore the real issue of studying $T_{G}(r,s)=0$ is when $f(k)\geq 2$, so there is no loss of generality in assuming $f(k)\geq 2$. Since all $g_{k,\ n/k}^{r,s}\geq 1$, we have $T_{G}(r,s)\geq 0$ and it is enough to show $T_{G}(r,s)\leq 0$ in order to conclude that $T_{G}(r,s)=0$. Now

[TABLE]

implies

[TABLE]

and so

[TABLE]

which gives what we claimed $T_{G}(r,s)=0$. On the othe hand, $G$ is a $\mathcal{Q}$-group because $f(k)\leq k$ and so it is solvable by (i) of Theorem 2.2. It follows that there exist $(r,s)\in\mathbb{R}^{2}$ such that $T_{G}(r,s)=0$ and $G$ is solvable. ∎

While the existence of $(r,s)$ is shown under the conditions of Theorem 2.3, the structural conditions of $G$ are elucidated in Theorem 2.2.