Abstract
We say that a semigroup S is (fixed-point-free, for short f.p.f.) permutable, if, for some integer n and for every x1,..., xn in S, there exists a non-trivial (fixed-point-free) permutation σ on {1,..., n}, such that:
In this paper we present the results of a systematical study of fixed-point-free permutation, pointing out the main differences from the well-known permutation property for semigroup and groups (to supply an interesting example, we study into details the f.p.f. permutation property in dihedral groups). We prove, by a direct combinatorial argument, that a finitely generated periodic and fixed-point-free permutable semigroup is finite and show that a f.p.f. permutable semigroup is finite if and only if its idempotents are central. Also, f.p.f. permutable groups are characterized: a group (resp. a finitely generated group) G is f.p.f. permutable if and only if its commutator subgroup G′ is finite (resp. G is center-by-finite). Finally, we define the weak f.p.f. permutability, which is proved to be equivalent to f.p.f. permutability for groups, but not for semigroups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. B. Blyth,Rewriting products of group elements, I, J. Algebra,116 (1988), pp. 506–521.
R. D.Blyth - D. J.Robinson,Recent Progress in Rewritability in Groups, Proc. Singapore Group Theory Conference (to appear).
M. Curzio -P. Longobardi -M. Maj,Su di un problema combinatorio in teoria dei gruppi, Atti Acc. Lindei Rend. fis., VIII,74 (1983), pp. 136–142.
M. Curzio -P. Longobardi -M. Maj -D. J. S. Robinson,A permutational property of groups, Arch. Math.,44 (1985), pp. 385–389.
A. De Luca -S. Varricchio,Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups, Theor. Comp. Sc.,63 (1989), pp. 333–348.
M. Garzon -Y. Zalcstein,On permutation properties in groups and semigroups, Semigroup Forum,35 (1987), pp. 337–351.
M. Garzon -Y. Zalcstein,Linear semigroups with permutation properties, Semigroup Forum,35 (1987), pp. 369–371.
J. Justin -G. Pirillo,On a natural extension of Jacob's rank, J. Comb. Th. Series A,43 (1986), pp. 205–218.
J. Justin -G. Pirillo,Comments on the permutation property for semigroups, Semigroup Forum,39 (1989), pp. 109–112.
J. Justin -G. Pirillo,Infinite words and permutation properties, Semigroup Forum,40 (1990), pp. 13–22.
G. Pirillo,On permutation property for semigroups, Group Theory conference (Bressanone/Brixen, 1986), Lectures Notes in Math., 1281, Springer, Berlin, 1987.
G. Pirillo,On permutation properties for finitely generated semigroups, Combinatorics '86 (Passo della Mendola, 1986), Ann. Discrete Math.,37, North-Holland, Amsterdam, 1988.
B. Piochi,Permutability of normal extensions of a semilattice, Semigroup Forum,43 (1991), pp. 151–162.
B. Piochi,Permutability of center-by-finite groups, Att. Acc. Lincei Rend. fis. (8), LXXXIII (1989), pp. 153–158.
B. Piochi -G. Pirillo,Sur une propriété de permutabilité des groupes finis, C. R. Acad. Sci. Paris,307-I (1988), pp. 115–117.
A.Postiglione - A.Rispoli,Una nota sulla proprietà di permutabilità, Rapp. Interno Dip. Informatica e Appl. Univ. Salerno,1 (1987), to appear in Atti Accademia Fisiocritici (Siena).
F. P. Ramsey,On a problem of formal logic, Proc. London Math. Soc. (2),30 (1930), pp. 264–286.
A. Restivo,Permutation properties and the Fibonacci semigroup, Semigroup Forum,38 (1989), pp. 337–345.
A. Restivo -C. Reutenauer,On the Burnside problem for semigroups, J. Algebra,89 (1984), pp. 102–104.
D. J. S. Robinson,Finiteness conditions and generalized soluble groups, Springer-Verlag, Berlin, Heidelberg, 1972.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Justin, J., Piochi, B. & Pirillo, G. On fixed-point-free permutation properties in groups and semigroups. Annali di Matematica pura ed applicata 159, 45–64 (1991). https://doi.org/10.1007/BF01766292
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01766292