Dense chains in the subgroup lattice\\[3pt] of a $\kappa$-existentially closed group
We answer a question of Brescia, Ersoy and Kuzucuoglu concerning the subgroup lattice of a $\kappa$-existentially closed group. We prove, for every infinite cardinal $\kappa$, that such a group contains an elementary abelian $2$-subgroup of rank $\kappa$. We then construct an explicit ordered set of cardinality $\kappa$ without endpoints, every nonempty open interval of which has cardinality $\kappa$. Indexing a basis of the elementary abelian subgroup by this set gives a strictly increasing chain of subgroups with exactly $\kappa$ members between any two distinct members. No regularity assumption on $\kappa$ is required.
Evidence-backed creditSubstantive mathematical contributions
Show proposalsConcrete proposals are tied to an immutable base version. When the paper owner confirms a contribution, it can be linked to the later manuscript version that incorporated it, was revised because of it, or resolved the issue independently. If the owner has not decided within 90 days, an administrator or a Managing Editor of the paper's subject area can take an editorial decision. Proposals by co-authors are recorded but give no contribution credit.
Substantive mathematical contributions
Show proposalsConcrete proposals are tied to an immutable base version. When the paper owner confirms a contribution, it can be linked to the later manuscript version that incorporated it, was revised because of it, or resolved the issue independently. If the owner has not decided within 90 days, an administrator or a Managing Editor of the paper's subject area can take an editorial decision. Proposals by co-authors are recorded but give no contribution credit.
Lemma 2.3
For every infinite cardinal $\kappa$ there is a linear order $(I,<)$ of cardinality $\kappa$, with no endpoints, such that \[ |\{u\in I:s<u<t\}|=\kappa\qquad(s<t). \]
Open the parent paper · This indexed statement is a discovery surface and does not itself imply human verification.
Mathematical dependencies
Click a node to select the same theorem, lemma or proposition in the review sidebar. The sidebar stays available while you inspect the graph. Solid arrows are explicit LaTeX references; dashed arrows are legacy structural suggestions and are never treated as proof evidence.
Versions
Every version is immutable. Open an earlier version or compare revisions here without leaving the paper reader layout.
References
ProofForum parses references embedded in the uploaded LaTeX. Internal PF IDs are linked and colored by the current human-review state of the cited paper: green = fully certified tracked structures, yellow = partial review/concern, red = incorrect point reported, grey = no human verification or unresolved internal ID. arXiv and Hexagon identifiers cited with a prefix or link are matched to ProofForum verification records: a state is shown only for the exact cited version; a citation without a version is marked as a matching work.
- External referenceM. Brescia, K. Ersoy and M. Kuzucuoglu, $\kappa$-existentially closed groups and centralizers, J. Algebra 679 (2025), 37--55. https://doi.org/10.1016/j.jalgebra.2025.05.013.LaTeX source\bibitem{BEK} M. Brescia, K. Ersoy and M. Kuzucuoglu, \emph{$\kappa$-existentially closed groups and centralizers}, J. Algebra \textbf{679} (2025), 37--55. \url{https://doi.org/10.1016/j.jalgebra.2025.05.013}.
- External referenceG. Higman and E. Scott, Existentially Closed Groups, London Mathematical Society Monographs, New Series, vol. 3, Clarendon Press, Oxford, 1988.LaTeX source\bibitem{HigmanScott} G. Higman and E. Scott, \emph{Existentially Closed Groups}, London Mathematical Society Monographs, New Series, vol.~3, Clarendon Press, Oxford, 1988.
Cited by
Published ProofForum papers that cite this paper or an exact persistent object in this immutable version.
Lean / Palomar
Machine-checked Lean evidence is linked through immutable Palomar registry records. ProofForum separately tracks statement-to-declaration correspondence and independent human alignment review.
Machine-readable theorem index · 4 statements
Public text index for scholarly discovery. It mirrors theorem, lemma, proposition and related statements from this immutable version and does not imply human verification.
- PF:2026.000019/v1/LEM-2.1
Let $G$ be $\kappa$-existentially closed, and let $E\leq G$ be an elementary abelian $2$-subgroup generated by fewer than $\kappa$ elements. There is an involution $x\in C_G(E)\setminus E$. Consequently, $\langle E,x\rangle=E\times\langle x\rangle$.
- PF:2026.000019/v1/LEM-2.2
Every $\kappa$-existentially closed group contains an elementary abelian $2$-subgroup of rank $\kappa$.
- PF:2026.000019/v1/LEM-2.3
For every infinite cardinal $\kappa$ there is a linear order $(I,<)$ of cardinality $\kappa$, with no endpoints, such that \[ |\{u\in I:s<u<t\}|=\kappa\qquad(s<t). \]
- PF:2026.000019/v1/THM-3.1
Let $\kappa$ be any infinite cardinal and let $G$ be a $\kappa$-existentially closed group. Then $G$ has a chain $\mathcal C$ of subgroups such that $|\mathcal C|=\kappa$, $\mathcal C$ has no least or greatest member, and \[ |\{K\in\mathcal C:H<K<L\}|=\kappa \quad\text{whenever } H,L\in\mathcal C\text{ and }H<L. \] In particular Problem reference has an affirmative answer.