# $\mathcal{Q}$-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.

## Key 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.