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Group closures of one-to-one transformations

Published online by Cambridge University Press:  17 April 2009

Inessa Levi
Inessa Levi
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY 40292, United States of America
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For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 177 - 188
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Fitzpatrick, S.P. and Symons, J.S.V., ‘Automorphisms of transformation semigroups’, Proc. Edinburgh Math. Soc. 19 (1974/1975), 327–329.CrossRefGoogle Scholar
[2]Levi, I., Schein, B.M., Sullivan, R.P. and Wood, G.R., ‘Automorphisms of Baer-Levi semigroups’, J. London Math. Soc. 28 (1983), 492–495.CrossRefGoogle Scholar
[3]Levi, I., ‘Automorphisms of normal transformation semigroups’, Proc. Edinburgh Math. Soc. 28 (1985), 185–205.CrossRefGoogle Scholar
[4]Levi, I., ‘Automorphisms of normal partial transformation semigroups’, Glasgow Math. J. 29 (1987), 149–157.CrossRefGoogle Scholar
[5]Levi, I., ‘Normal semigroups of one-to-one transformations’, Proc. Edinburgh Math. Soc. 34 (1991), 65–76.CrossRefGoogle Scholar
[6]Levi, I., ‘On the inner automorphisms of finite transformation semigroups’, Proc. Edinburgh Math. Soc. 30 (1996), 27–30.CrossRefGoogle Scholar
[7]Levi, I., ‘On groups associated with transformation semigroups’, Semigroup Forum 59 (1999), 1–12.CrossRefGoogle Scholar
[8]Levi, I., McAlister, D.B. and McFadden, R.B., ‘n - normal semigroups’, Semigroup Forum 62 (2001), 173–177.CrossRefGoogle Scholar
[9]Levi, I., McAlister, D.B. and McFadden, R.B., ‘Groups associated with finite transformation semigroups’, Semigroup Forum 61 (2000), 453–467.CrossRefGoogle Scholar
[10]Levi, I. and Seif, S., ‘Finite normal semigroups’, Semigroup Forum 57 (1998), 69–74.CrossRefGoogle Scholar
[11]Levi, I. and McFadden, R.B., ‘Fully invariant transformations and associated groups’, Comm. Algebra 28 (2000), 4829–4838.CrossRefGoogle Scholar
[12]Levi, I. and Wood, J., ‘Group closures of partial transformations’, (submitted).Google Scholar
[13]Scott, W.R., Group theory (Prentice Hall, N.J., 1964).Google Scholar
[14]Sullivan, R.P., ‘Automorphisms of transformation semigroups’, J. Austral. Math. Soc. 20 (1975), 77–84.CrossRefGoogle Scholar
[15]Sullivan, R.P., ‘Automorphisms of injective transformation semigroups’, Studia Sci. Math. Hungar. 15 (1980), 1–4.Google Scholar
[16]Sutov, E.G., ‘On semigroups of almost identical transformations’, Soviet Math. Dokl. 1 (1960), 1080–1083.Google Scholar
[17]Sutov, E.G., ‘Homomorphisms of the semigroup of all partial transformations’, Izv. Vysšs. Učebn. Zaved. Mat. 22 (1962), 177–184.Google Scholar
[18]Symons, J.S.V., ‘Normal transformation semigroups’, J. Austral. Math. Soc. Ser. A 22 (1976), 385–390.CrossRefGoogle Scholar