Invariant systems of representatives, or the cost of symmetry. (English) Zbl 1476.05070
Main Theorem. Suppose that a group \(G\) acts on a set \(U\), and \(\mathcal F\) is a \(G\)-invariant family of finite subsets of \(U\) of uniformly bounded cardinality. Let \(X \subseteq U\) be a finite system of representatives for this family i.e. \(X \cap F \not= \emptyset\) for any \(F \in \mathcal F\). Then there exists a \(G\)-invariant system of representatives \(Y\) such that \(|Y| \leq |X| \cdot \max_{F\in \mathcal F} |F |\).
The proof of the main theorem is elementary, except that the authors use B. Neumann’s theorem on covering groups by cosets.
Corollary. Let \(\Gamma\) be a graph and let \(K\) be a finite graph. Then if \(\Gamma\) contains a finite set of vertices (edges) \(X\) such that each subgraph of \(\Gamma\) isomorphic to \(K\) has at least one vertex (edge) from \(X\), then \(\Gamma\) contains a finite set of vertices (edges) \(Y\), \(|Y|\leq |X|\cdot \) (the number of vertices (edges) of \(K\), invariant with respect to all automorphisms of \(\Gamma\) and such that again each subgraph of \(\Gamma\) isomorphic to \(K\) has at least one vertex (edge) from \(Y\).
The proof of the main theorem is elementary, except that the authors use B. Neumann’s theorem on covering groups by cosets.
Corollary. Let \(\Gamma\) be a graph and let \(K\) be a finite graph. Then if \(\Gamma\) contains a finite set of vertices (edges) \(X\) such that each subgraph of \(\Gamma\) isomorphic to \(K\) has at least one vertex (edge) from \(X\), then \(\Gamma\) contains a finite set of vertices (edges) \(Y\), \(|Y|\leq |X|\cdot \) (the number of vertices (edges) of \(K\), invariant with respect to all automorphisms of \(\Gamma\) and such that again each subgraph of \(\Gamma\) isomorphic to \(K\) has at least one vertex (edge) from \(Y\).
Reviewer: V. A. Roman’kov (Omsk)
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05C69 | Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) |
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
References:
| [1] | Bruno, B.; Napolitani, F., A note on nilpotent-by-Černikov groups, Glasg. Math. J., 46, 211-215 (2004) · Zbl 1056.20022 |
| [2] | Chermak, A.; Delgado, A., A measuring argument for finite group, Proc. Amer. Math. Soc., 107, 907-914 (1989) · Zbl 0687.20022 |
| [3] | de Giovanni, F.; Trombetti, M., Large characteristic subgroups with modular subgroup lattice, Arch. Math, 111, 2, 123-128 (2018) · Zbl 1448.20028 |
| [4] | de Giovanni, F.; Trombetti, M., A note on large characteristic subgroups, Comm. Algebra, 46, 11, 4654-4662 (2018) · Zbl 1397.20035 |
| [5] | de Giovanni, F.; Trombetti, M., Large characteristic subgroups in which normality is a transitive relation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., 30, 255-268 (2019) · Zbl 1453.20038 |
| [6] | de Giovanni, F.; Trombetti, M., Large characteristic subgroups and abstract group classes, Quaest. Math., 43, 8, 1159-1172 (2020) · Zbl 1457.20024 |
| [7] | E. Frolova, Khukhro-Makarenko type theorems for algebras, arXiv:1804.00268. |
| [8] | Isaacs, I. M., Finite Group Theory (2008), American Math. Soc.: American Math. Soc. Providence RI · Zbl 1169.20001 |
| [9] | Kargapolov, M. I.; Merzljakov, Yu. I., (Fundamentals of the Theory of Groups. Fundamentals of the Theory of Groups, Graduate Texts in Mathematics, vol. 62 (1979), Springer) · Zbl 0549.20001 |
| [10] | Khukhro, E. I.; Klyachko, Ant. A.; Makarenko, N. Yu.; Melnikova, Yu. B., Automorphism invariance and identities, Bull. Lond. Math. Soc., 41, 5, 804-816 (2009), arXiv:0812.1359 · Zbl 1250.20023 |
| [11] | Khukhro, E. I.; Yu. Makarenko, N., Characteristic nilpotent subgroups of bounded co-rank and automorphically-invariant ideals of bounded codimension in Lie algebras, Q. J. Math., 58, 229-247 (2007) · Zbl 1147.17007 |
| [12] | Khukhro, E. I.; Yu. Makarenko, N., Large characteristic subgroups satisfying multilinear commutator identities, J. Lond. Math. Soc., 75, 3, 635-646 (2007) · Zbl 1132.20013 |
| [13] | Khukhro, E. I.; Yu. Makarenko, N., Automorphically-invariant ideals satisfying multilinear identities, and group-theoretic applications, J. Algebra, 320, 4, 1723-1740 (2008) · Zbl 1163.17002 |
| [14] | Klyachko, A. A.; Mel’nikova, Yu. B., A short proof of the Khukhro-Makarenko theorem on large characteristic subgroups with laws, Sb. Math., 200, 5, 661-664 (2009), arXiv:0805.2747 · Zbl 1191.20030 |
| [15] | Klyachko, A. A.; Milentyeva, M. V., Large and symmetric: The Khukhro-Makarenko theorem on laws — without laws, J. Algebra, 424, 222-241 (2015), arXiv:1309.0571 · Zbl 1337.20028 |
| [16] | Makarenko, N. Yu.; Shumyatsky, P., Characteristic subgroups in locally finite groups, J. Algebra, 352, 1, 354-360 (2012) · Zbl 1255.20038 |
| [17] | Neumann, B. H., Groups covered by permutable subsets, J. Lond. Math. Soc., s1-29, 2, 236-248 (1954) · Zbl 0055.01604 |
| [18] | Podoski, K.; Szegedy, B., Bounds in groups with finite abelian coverings or with finite derived groups, J. Group Theory, 5, 4, 443-452 (2002) · Zbl 1018.20024 |
| [19] | Vdovin, E. P., Large normal nilpotent subgroups of finite groups, Sib. Math. J., 41, 2, 246-251 (2000) · Zbl 0956.20008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
