Home Mathematics Some necessary conditions for compatibility of groups
Article
Licensed
Unlicensed Requires Authentication

Some necessary conditions for compatibility of groups

  • Zhaochen Ding ORCID logo EMAIL logo and Gabriel Verret
Published/Copyright: January 17, 2026
Journal of Group Theory
From the journal Journal of Group Theory

Abstract

Two groups L 1 and L 2 are compatible if there exists a finite group 𝐺 with isomorphic normal subgroups N 1 and N 2 such that L 1 G / N 1 and L 2 G / N 2 . In this paper, we give new necessary conditions for two groups to be compatible.

Funding source: Marsden Fund

Award Identifier / Grant number: UOA1824

Funding statement: The authors gratefully acknowledge the financial support of the Marsden Fund UOA1824 and the University of Auckland.

Acknowledgements

We thank the anonymous referee for their helpful comments.

  1. Communicated by: Michael Giudici

References

[1] P. J. Cameron, Permutation groups with multiply transitive suborbits, Proc. Lond. Math. Soc. (3) 25 (1972), 427–440. 10.1112/plms/s3-25.3.427Search in Google Scholar

[2] J. B. Fawcett, M. Giudici, C. H. Li, C. E. Praeger, G. Royle and G. Verret, Primitive permutation groups with a suborbit of length 5 and vertex-primitive graphs of valency 5, J. Combin. Theory Ser. A 157 (2018), 247–266. 10.1016/j.jcta.2018.02.008Search in Google Scholar

[3] M. Giudici, S. P. Glasby, C. H. Li and G. Verret, Arc-transitive digraphs with quasiprimitive local actions, J. Pure Appl. Algebra 223 (2019), no. 3, 1217–1226. 10.1016/j.jpaa.2018.06.002Search in Google Scholar

[4] D. M. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. (2) 111 (1980), no. 2, 377–406. 10.2307/1971203Search in Google Scholar

[5] E. I. Khukhro and V. D. Mazurov, The Kourovka Notebook. Unsolved Problems in Group Theory, Sobolev Institute of Mathematics, Novosibirsk, 2022. Search in Google Scholar

[6] W. Knapp, On the point stabilizer in a primitive permutation group, Math. Z. 133 (1973), 137–168. 10.1007/BF01237901Search in Google Scholar

[7] H. Kurzweil and B. Stellmacher, The Theory of Finite Groups, Universitext, Springer, New York, 2004. 10.1007/b97433Search in Google Scholar

[8] C. H. Li, Z. P. Lu and D. Marušič, On primitive permutation groups with small suborbits and their orbital graphs, J. Algebra 279 (2004), no. 2, 749–770. 10.1016/j.jalgebra.2004.03.005Search in Google Scholar

[9] W. L. Quirin, Primitive permutation groups with small orbitals, Math. Z. 122 (1971), 267–274. 10.1007/BF01109920Search in Google Scholar

[10] C. C. Sims, Graphs and finite permutation groups, Math. Z. 95 (1967), 76–86. 10.1007/BF01117534Search in Google Scholar

[11] R. M. Weiss, Über symmetrische Graphen, deren Valenz eine Primzahl ist, Math. Z. 136 (1974), 277–278. 10.1007/BF01214131Search in Google Scholar

[12] H. Wielandt, Eine Verallgemeinerung der invarianten Untergruppen, Math. Z. 45 (1939), no. 1, 209–244. 10.1007/BF01580283Search in Google Scholar

Received: 2025-01-23
Revised: 2025-12-02
Published Online: 2026-01-17

© 2026 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 30.1.2026 from https://www.degruyterbrill.com/document/doi/10.1515/jgth-2025-0013/html?lang=en
Scroll to top button