Normal divisors of a 2-transitive group of automorphisms of a linearly ordered set

E. B. Rabinovich   ; V. Z. Feinberg

Sbornik. Mathematics, Tome 22 (1974) no. 2, pp. 187-200

Résumé

The main result of the paper is a description of the normal structure of the groups $\operatorname{Aut}(X,\leqslant)$, where $X$ is a linearly ordered set satisfying one of the following equivalent conditions: I. $\operatorname{Aut}(X,\leqslant)$ is 2-transitive. II. $\operatorname{Aut}(X,\leqslant)$ is $k$-transitive. III. $X$ does not have a greatest or a least element, and any two intervals $[a,b]$, $a<b$ and $[c,d]$, $c<d$, are similar. IV. $\operatorname{Aut}(X,\leqslant)$ is a 0-primitive, transitive, nonregular permutation group. Main theorem. {\it Suppose $\operatorname{Aut}(X,\leqslant)$ is $2$-transitive. Then $\overline A,$ $\overset\rightarrow A$ and $\overset\leftarrow A$ are the only nontrivial normal and subnormal subgroups of $\operatorname{Aut}(X,\leqslant)$. Here \begin{gather*} \overset\leftarrow A=\{g\in\operatorname{Aut}(X,\leqslant)\mid \operatorname{Tr}g\text{ is bounded below}\},\\ \overset\rightarrow A=\{g\in\operatorname{Aut}(X,\leqslant)\mid \operatorname{Tr}g\text{ is bounded above}\},\\ \overline A=\overset\rightarrow A\cap\overset\leftarrow A,\qquad\operatorname{Tr}g=\{x\in X\mid g(x)\ne x\}. \end{gather*}} Bibliography: 21 titles.

Bibliographie

[1] I. Schreier, S. Ulam, “Über die Permututionsgruppe der natürlichen Zahlenfolge”, Studia Math., 4 (1933), 134–141 | Zbl

[2] R. Baer, “Die Kompositionsreihe der Gruppe aller einenideutigen Abbildungen einer unenddlichen Menge auf sich”, Studia Math., 5 (1934), 15–17 | Zbl

[3] B. I. Plotkin, Gruppy avtomorfizmov algebraicheskikh sistem, izd-vo «Nauka», Moskva, 1966 | MR

[4] G. Higman, “On infinite simple permutation groups”, Publ. Math., 3:3–4 (1954), 221–226 | MR | Zbl

[5] H. Willandt, “Gedanken für eine allgemeine Theorie der Permutationsgruppen”, Rend. Semin. Math. Univ. Torino, 21 (1961/62), 31–39 | MR

[6] L. Fuks, Chastichno uporyadochennye algebraicheskie sistemy, IL, Moskva, 1965 | MR

[7] N. Burbaki, Obschaya topologiya. Topologicheskie gruppy. Chisla i svyazannye s nimi gruppy i prostranstva, izd-vo «Nauka», Moskva, 1969 | MR

[8] C. Holland, “A class of simple lattice-ordered groups”, Proc. Amer. Math. Soc., 16:2 (1965), 326–329 | DOI | MR | Zbl

[9] E. C. Weinberg, “Embedding in a divisible lattice-ordered group”, J. London Math. Soc., 42:3 (1967), 504–506 | DOI | MR | Zbl

[10] H. Wielandt, Unendliche Permutationsgruppen, Tübingen, 1960

[11] J. T. Lloyd, Lattice-ordered groups and O-permutation groups, Tulane University, 1964

[12] C. Holland, “Transitive lattice-ordered permutation groups”, Math. Z., 87:3 (1965), 420–433 | DOI | MR | Zbl

[13] S. H. McCleary, “0-primitive ordered permutation groups”, Pacific J. Math., 40:2 (1972), 349–372 | MR | Zbl

[14] L. B. Treybig, “Conserning homogeneity in totally ordered connected topological space”, Pacific J. Math., 13:4 (1963), 1417–1421 | MR | Zbl

[15] T. Ohkuma, “Sur quelques ensembles ordonnés lineáirment”, Fundam. math., 43 (1964), 326–337 | MR

[16] K. Kuratovskii, A. Mostovskii, Teoriya mnozhestv, izd-vo «Mir», Moskva, 1970 | MR

[17] C. Holland, “The lattice-ordered group of automorphisms of an ordered set”, Michigan Math. J., 10:4 (1963), 399–409 | DOI | MR

[18] R. D. Anderson, “The algebraic simplicity of certain groups of homeomorphisms”, Amer. J. Math., 80:4 (1958), 955–963 | DOI | MR | Zbl

[19] W. R. Scott, Group theory, Englewood Cliffs, Prentice-Hull, 1964 | MR | Zbl

[20] A. Karras, D. Solitar, “Some remarks of the infinite symmetric groups”, Math. Z., 66:1 (1956), 64–94 | DOI | MR

[21] D. Eisenbud, “Group of order automorphism of certain homogenous ordered sets”, Michigan Math. J., 16:1 (1969), 59–63 | DOI | MR | Zbl