Locally graded groups with dual classes of subgroups
We prove that torsion-free locally graded groups in which every subgroup is pronormal or contained in the pro-norm are abelian. We then show that every insoluble locally graded group with this property is locally finite. Thus the classification of Ingrosso, Trombetti and Zitouni holds without the hypothesis of local $c$-gradedness. The proof uses results on finitely generated HNN-free groups due to Robinson, Russo and Vincenzi.
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Substantive mathematical contributions
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Theorem 3.1
Let $G$ be a torsion-free locally graded group satisfying (equation reference). Then $G$ is abelian.
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- External referenceF. de Giovanni and M. Trombetti, Pronormality in group theory, Adv. Group Theory Appl. 9 (2020), 123--149. https://doi.org/10.32037/agta-2020-005.LaTeX source\bibitem{dGT} F. de Giovanni and M. Trombetti, \emph{Pronormality in group theory}, Adv. Group Theory Appl. \textbf{9} (2020), 123--149. \url{https://doi.org/10.32037/agta-2020-005}.
- External referenceE. Ingrosso, M. Trombetti and A. Zitouni, Groups with dual classes of subgroups, J. Aust. Math. Soc. (2026), First View, 1--40. https://doi.org/10.1017/S1446788726101682.LaTeX source\bibitem{ITZ} E. Ingrosso, M. Trombetti and A. Zitouni, \emph{Groups with dual classes of subgroups}, J. Aust. Math. Soc. (2026), First View, 1--40. \url{https://doi.org/10.1017/S1446788726101682}.
- External referenceD. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Springer, New York, 1996.LaTeX source\bibitem{Robinson} D. J. S. Robinson, \emph{A Course in the Theory of Groups}, 2nd ed., Springer, New York, 1996.
- External referenceD. J. S. Robinson, A. Russo and G. Vincenzi, On groups which contain no HNN-extensions, Internat. J. Algebra Comput. 17 (2007), 1377--1387. https://doi.org/10.1142/S0218196707004311.LaTeX source\bibitem{RRV} D. J. S. Robinson, A. Russo and G. Vincenzi, \emph{On groups which contain no HNN-extensions}, Internat. J. Algebra Comput. \textbf{17} (2007), 1377--1387. \url{https://doi.org/10.1142/S0218196707004311}.
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Machine-readable theorem index · 6 statements
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- PF:2026.000017/v1/LEM-2.1
Let $G$ be torsion-free, locally graded and satisfy (equation reference). Then every finite homomorphic image of $G$ is metabelian.
- PF:2026.000017/v1/THM-3.1
Let $G$ be a torsion-free locally graded group satisfying (equation reference). Then $G$ is abelian.
- PF:2026.000017/v1/COR-3.2
Every infinite simple locally graded group fails to satisfy (equation reference).
- PF:2026.000017/v1/LEM-4.1
Every locally graded group satisfying (equation reference) is HNN-free.
- PF:2026.000017/v1/LEM-4.2
Let $F$ be a finitely generated locally graded group satisfying (equation reference). Then $F$ has finite upper rank and is virtually polycyclic.
- PF:2026.000017/v1/THM-4.3
Let $G$ be an insoluble locally graded group. Then $G$ satisfies (equation reference) if and only if one of the following alternatives holds: \item $G=H\times K$, where $H$ is periodic and all its subgroups are pronormal, $\pi(H)\cap\pi(K)\subseteq\{2\}$, and the Sylow $2$-subgroups of $H$ are elementary abelian whenever $Z(K)\ne1$. Moreover, $K$ is isomorphic to one of the groups \[ J_1,\qquad \PSL(2,2^p)\quad(p\text{ prime}), \] or to $\PSL(2,q)$ or $\SL(2,q)$, where $q$ satisfies one of \[ q=5^m\quad(m=1\text{ or }m\text{ an odd prime}),\\ q>5,\quad q\equiv3,5\pmod8,\\ q=2^m\quad(m\text{ prime}). \] \item $G=C\ltimes(H\times K)$, where $K=\PSL(2,2^p)$ for a prime $p>3$, $|C|=p^n$ for some $n\geq1$, $[C,K]…