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.
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- 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.
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Machine-readable theorem index · 4 statements
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- 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.