Conjugacy classes whose square is an infinite symmetric group
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- Trans. Amer. Math. Soc. 316 (1989), 493-522 Request permission
Abstract:
Let ${X_\nu }$ be the set of all permutations $\xi$ of an infinite set $A$ of cardinality ${\aleph _\nu }$ with the property: every permutation of $A$ is a product of two conjugates of $\xi$. The set ${X_0}$ is shown to be the set of permutations $\xi$ satisfying one of the following three conditions: (1) $\xi$ has at least two infinite orbits. (2) $\xi$ has at least one infinite orbit and infinitely many orbits of a fixed finite size $n$. (3) $\xi$ has: no infinite orbit; infinitely many finite orbits of size $k,l$ and $k + l$ for some positive integers $k,l$; and infinitely many orbits of size $> 2$. It follows that $\xi \in {X_0}$ iff some transposition is a product of two conjugates of $\xi$, and $\xi$ is not a product $\sigma i$, where $\sigma$ has a finite support and $i$ is an involution. For $\nu > 0,\;\xi \in {X_\nu }$ iff $\xi$ moves ${\aleph _\nu }$ elements, and satisfies (1), (2) or $(3’)$, where $(3’)$ is obtained from (3) by omitting the requirement that $\xi$ has infinitely many orbits of size $> 2$. It follows that for $\nu > 0,\;\xi \in {X_\nu }\;$ iff $\xi$ moves ${\aleph _\nu }$ elements and some transposition is the product of two conjugates of $\xi$. The covering number of a subset $X$ of a group $G$ is the smallest power of $X$ (if any) that equals $G$ [AH]. These results complete the classification of conjugacy classes in infinite symmetric groups with respect to their covering number.References
-
Z. Arad, D. Chillag and G. Moran, Groups with a small covering number, Chapter 4 of [AH].
- Z. Arad and M. Herzog (eds.), Products of conjugacy classes in groups, Lecture Notes in Mathematics, vol. 1112, Springer-Verlag, Berlin, 1985. MR 783067, DOI 10.1007/BFb0072284
- Edward A. Bertram, On a theorem of Schreier and Ulam for countable permutations, J. Algebra 24 (1973), 316–322. MR 308276, DOI 10.1016/0021-8693(73)90141-5
- G. Boccara, Cycles comme produit de deux permutations de classes données, Discrete Math. 38 (1982), no. 2-3, 129–142 (French). MR 676530, DOI 10.1016/0012-365X(82)90282-5
- Manfred Droste, Products of conjugacy classes of the infinite symmetric groups, Discrete Math. 47 (1983), no. 1, 35–48. MR 720606, DOI 10.1016/0012-365X(83)90070-5
- Manfred Droste, Cubes of conjugacy classes covering the infinite symmetric group, Trans. Amer. Math. Soc. 288 (1985), no. 1, 381–393. MR 773066, DOI 10.1090/S0002-9947-1985-0773066-3
- Manfred Droste, Squares of conjugacy classes in the infinite symmetric groups, Trans. Amer. Math. Soc. 303 (1987), no. 2, 503–515. MR 902781, DOI 10.1090/S0002-9947-1987-0902781-5
- Manfred Droste and Rüdiger Göbel, On a theorem of Baer, Schreier, and Ulam for permutations, J. Algebra 58 (1979), no. 2, 282–290. MR 540639, DOI 10.1016/0021-8693(79)90161-3
- Manfred Droste and Rüdiger Göbel, Products of conjugate permutations, Pacific J. Math. 94 (1981), no. 1, 47–60. MR 625807 Y. Dvir, Covering properties of permutation groups, Chapter 4 of [AH]. A. B. Gray, Infinite symmetric and monomial groups, Ph.D. Thesis, New Mexico State University, Las Cruces, New Mexico, 1960.
- P. Hall, Some constructions for locally finite groups, J. London Math. Soc. 34 (1959), 305–319. MR 162845, DOI 10.1112/jlms/s1-34.3.305 G. Moran, The algebra of reflections of an infinite set, Notices Amer. Math. Soc. 73T (1973), A193.
- Gadi Moran, The product of two reflection classes of the symmetric group, Discrete Math. 15 (1976), no. 1, 63–77. MR 412297, DOI 10.1016/0012-365X(76)90110-2
- Gadi Moran, Parity features for classes of the infinite symmetric group, J. Combin. Theory Ser. A 33 (1982), no. 1, 82–98. MR 665658, DOI 10.1016/0097-3165(82)90081-4
- Gadi Moran, Of planar Eulerian graphs and permutations, Trans. Amer. Math. Soc. 287 (1985), no. 1, 323–341. MR 766222, DOI 10.1090/S0002-9947-1985-0766222-1
- Gadi Moran, The products of conjugacy classes in some infinite simple groups, Israel J. Math. 50 (1985), no. 1-2, 54–74. MR 788069, DOI 10.1007/BF02761118
- Gadi Moran, Products of involution classes in infinite symmetric groups, Trans. Amer. Math. Soc. 307 (1988), no. 2, 745–762. MR 940225, DOI 10.1090/S0002-9947-1988-0940225-9
- Oystein Ore, Some remarks on commutators, Proc. Amer. Math. Soc. 2 (1951), 307–314. MR 40298, DOI 10.1090/S0002-9939-1951-0040298-4
- W. R. Scott, Group theory, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0167513
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 316 (1989), 493-522
- MSC: Primary 20B07
- DOI: https://doi.org/10.1090/S0002-9947-1989-1020501-5
- MathSciNet review: 1020501