Products of involution classes in infinite symmetric groups

Author:
Gadi Moran

Journal:
Trans. Amer. Math. Soc. **307** (1988), 745-762

MSC:
Primary 20B30

DOI:
https://doi.org/10.1090/S0002-9947-1988-0940225-9

MathSciNet review:
940225

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an infinite set. Denote by the group of all permutations of , and let , denote the class of involutions of moving elements and fixing elements . The products were determined in [**M1**]. In this article we treat the products for . Let INF denote the set of permutations in moving infinitely many elements. We show:

(1) for .

(2)(a) if contains two integers of different parity; (b) if and all integers in have the same parity.

(3) , where iff satisfies one of the following three conditions:

(i) moves precisely three elements.

(ii) moves precisely five elements.

(iii) moves precisely seven elements and has order .

These results were announced in 1973 in [**MO**]. (1) and part of (2)(a) were generalized recently by Droste [**D1, D2**].

**[B]**E. Bertram,*On a theorem of Schreier and Ulam for countable permutations*, J. Algebra**24**(1973), 316-322. MR**0308276 (46:7390)****[Bo]**G. Boccara,*Cycles comme produit de deux permutations de classes donnees*, Discrete Math.**38**(1982), 129-142. MR**676530 (84e:20007)****[D1]**M. Droste,*Products of conjugacy classes of the infinite symmetric groups*, Discrete Math.**47**(1983), 35-48. MR**720606 (85b:20002)****[D2]**-,*Cubes of conjugacy classes covering the infinite symmetric groups*, Trans. Amer. Math. Soc.**288**(1985), 381-393. MR**773066 (86g:20003)****[D3]**-,*Classes of universal words for the infinite symmetric groups*, Algebra Universalis**20**(1985), 205-216. MR**806615 (87d:20049)****[DG]**M. Droste and R. Göbel,*On a theorem of Baer, Schreier and Ulam for permutations*, J. Algebra**58**(1979), 282-290. MR**540639 (80g:20008)****[Dv]**Y. Dvir,*Covering properties of permutation groups*, Products of Conjugacy Classes in Groups (Z. Arad and M. Herzog, eds.), Lecture Notes in Math., vol. 1112, Springer-Verlag, 1985. MR**783067 (87h:20001)****[G]**A. B. Gray,*Infinite symmetric and monomial groups*, Ph.D. Thesis, New Mexico State Univ., Las Cruces, N.M., 1960.**[MO]**G. Moran,*The algebra of reflections of an infinite set*, Notices Amer. Math. Soc.**73T**(1973), A193.**[M1]**-,*The product of two reflection classes of the symmetric group*, Discrete Math.**15**(1976), 63-77. MR**0412297 (54:423)****[M2]**-,*Reflection classes whose cubes cover the alternating group*, J. Combin. Theory Ser. A**21**(1976), 1-19. MR**0414673 (54:2769)****[M3]**-,*Permutations as products of**conjugate involutions*, J. Combin. Theory Ser. A**19**(1975), 240-242. MR**0379635 (52:540)****[M4]**-,*Trees and the bireflection property*, Israel J. Math.**41**(1982), 244-260. MR**657859 (83j:05030)****[M5]**-,*Of planar Eulerian graphs and permutations*, Trans. Amer. Math. Soc.**287**(1985), 323-341. MR**766222 (86i:05081)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
20B30

Retrieve articles in all journals with MSC: 20B30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0940225-9

Article copyright:
© Copyright 1988
American Mathematical Society