Conjugacy classes whose square is an infinite symmetric group

Author:
Gadi Moran

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

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Abstract: Let be the set of all permutations of an infinite set of cardinality with the property: every permutation of is a product of two conjugates of . The set is shown to be the set of permutations satisfying one of the following three conditions:

(1) has at least two infinite orbits.

(2) has at least one infinite orbit and infinitely many orbits of a fixed finite size .

(3) has: no infinite orbit; infinitely many finite orbits of size and for some positive integers ; and infinitely many orbits of size . It follows that iff some transposition is a product of two conjugates of , and is not a product , where has a finite support and is an involution.

For iff moves elements, and satisfies (1), (2) or , where is obtained from (3) by omitting the requirement that has infinitely many orbits of size . It follows that for iff moves elements and some transposition is the product of two conjugates of .

The covering number of a subset of a group is the smallest power of (if any) that equals [AH]. These results complete the classification of conjugacy classes in infinite symmetric groups with respect to their covering number.

**[ACM]**Z. Arad, D. Chillag and G. Moran,*Groups with a small covering number*, Chapter 4 of [AH].**[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****[B]**Edward A. Bertram,*On a theorem of Schreier and Ulam for countable permutations*, J. Algebra**24**(1973), 316–322. MR**0308276**, https://doi.org/10.1016/0021-8693(73)90141-5**[Bo]**G. Boccara,*Cycles comme produit de deux permutations de classes données*, Discrete Math.**38**(1982), no. 2-3, 129–142 (French). MR**676530**, https://doi.org/10.1016/0012-365X(82)90282-5**[D1]**Manfred Droste,*Products of conjugacy classes of the infinite symmetric groups*, Discrete Math.**47**(1983), no. 1, 35–48. MR**720606**, https://doi.org/10.1016/0012-365X(83)90070-5**[D2]**Manfred Droste,*Cubes of conjugacy classes covering the infinite symmetric group*, Trans. Amer. Math. Soc.**288**(1985), no. 1, 381–393. MR**773066**, https://doi.org/10.1090/S0002-9947-1985-0773066-3**[D3]**Manfred Droste,*Squares of conjugacy classes in the infinite symmetric groups*, Trans. Amer. Math. Soc.**303**(1987), no. 2, 503–515. MR**902781**, https://doi.org/10.1090/S0002-9947-1987-0902781-5**[DG1]**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**, https://doi.org/10.1016/0021-8693(79)90161-3**[DG2]**Manfred Droste and Rüdiger Göbel,*Products of conjugate permutations*, Pacific J. Math.**94**(1981), no. 1, 47–60. MR**625807****[Dv]**Y. Dvir,*Covering properties of permutation groups*, Chapter 4 of [AH].**[G]**A. B. Gray,*Infinite symmetric and monomial groups*, Ph.D. Thesis, New Mexico State University, Las Cruces, New Mexico, 1960.**[H]**P. Hall,*Some constructions for locally finite groups*, J. London Math. Soc.**34**(1959), 305–319. MR**0162845**, https://doi.org/10.1112/jlms/s1-34.3.305**[M1]**G. Moran,*The algebra of reflections of an infinite set*, Notices Amer. Math. Soc.**73T**(1973), A193.**[M2]**Gadi Moran,*The product of two reflection classes of the symmetric group*, Discrete Math.**15**(1976), no. 1, 63–77. MR**0412297**, https://doi.org/10.1016/0012-365X(76)90110-2**[M3]**Gadi Moran,*Parity features for classes of the infinite symmetric group*, J. Combin. Theory Ser. A**33**(1982), no. 1, 82–98. MR**665658**, https://doi.org/10.1016/0097-3165(82)90081-4**[M4]**Gadi Moran,*Of planar Eulerian graphs and permutations*, Trans. Amer. Math. Soc.**287**(1985), no. 1, 323–341. MR**766222**, https://doi.org/10.1090/S0002-9947-1985-0766222-1**[M5]**Gadi Moran,*The products of conjugacy classes in some infinite simple groups*, Israel J. Math.**50**(1985), no. 1-2, 54–74. MR**788069**, https://doi.org/10.1007/BF02761118**[M6]**Gadi Moran,*Products of involution classes in infinite symmetric groups*, Trans. Amer. Math. Soc.**307**(1988), no. 2, 745–762. MR**940225**, https://doi.org/10.1090/S0002-9947-1988-0940225-9**[O]**Oystein Ore,*Some remarks on commutators*, Proc. Amer. Math. Soc.**2**(1951), 307–314. MR**0040298**, https://doi.org/10.1090/S0002-9939-1951-0040298-4**[S]**W. R. Scott,*Group theory*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0167513**

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DOI:
https://doi.org/10.1090/S0002-9947-1989-1020501-5

Article copyright:
© Copyright 1989
American Mathematical Society