Locally graded groups with dual classes of subgroups
We prove that every torsion-free locally graded group in which each subgroup is either pronormal or contained in the pro-norm is abelian. Consequently, no infinite simple locally graded group has this property. We also extend the classification of insoluble locally $c$-graded groups with the same property to all locally graded groups. The proof combines results on pronormal subgroups with the theory of finitely generated HNN-free groups.
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Substantive mathematical contributions
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Lemma 4.2
Let $F$ be a finitely generated locally graded group satisfying (equation reference). Then $F$ has finite upper rank and is virtually polycyclic.
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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/v2/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/v2/THM-3.1
Let $G$ be a torsion-free locally graded group satisfying (equation reference). Then $G$ is abelian.
- PF:2026.000017/v2/COR-3.2
Every infinite simple locally graded group fails to satisfy (equation reference).
- PF:2026.000017/v2/LEM-4.1
Every locally graded group satisfying (equation reference) is HNN-free.
- PF:2026.000017/v2/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/v2/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]…