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Semigroups with few endomorphisms

In Memoriam Anatoli Ivanovič Mal'cev

Published online by Cambridge University Press:  09 April 2009

Vlastimil Dlab  and
B. H. Neumann
Vlastimil Dlab
Affiliation:
The Australian National UniversityCanberra
B. H. Neumann
Affiliation:
The Australian National UniversityCanberra
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Large finite groups have large automorphism groups [4]; infinite groups may, like the infinite cyclic group, have finite automorphism groups, but their endomorphism semigroups are infinite (see Baer [1, p. 530] or [2, p. 68]). We show in this paper that the corresponding propositions for semigroups are false.

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 1-2 , August 1969 , pp. 162 - 168
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Baer, Reinhold, ‘Finite extensions of abelian groups with minimum condition’, Trans. Amer. Math. Soc. 79, (1955) 521–540.CrossRefGoogle Scholar
[2]Baer, Reinhold, ‘Kriterien für die Endlichkeit von Gruppen’, Jber. Deutsch. Math.-Verein. 63, (1960) 53–77.Google Scholar
[3]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, volume II, (Amer. Math. Soc., Providence, R.I. 1967).CrossRefGoogle Scholar
[4]Ledermann, W. and Neumann, B. H., ‘On the order of the automorphism group of a finite group. I’, Proc. Roy. Soc. London, A, 233, (1956) 494–506.Google Scholar
[5]Neumann, B. H., ‘Some remarks on semigroup presentations’, Canad. J. Math. 19, (1967) 1018–1026.CrossRefGoogle Scholar