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When endomorphisms of G inducing automorphisms of G/V are automorphisms

Published online by Cambridge University Press:  20 January 2009

O. Macedońska
O. Macedońska
Affiliation:
Instytut Matematyki, Politechnika Ślaska, Zwyciestwa 42, 44-100 Gliwice, Poland
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Let G denote a relatively free group of a finite or countably infinite rank with a fixed set of free generators x1,x2,…,G′ the commutator subgroup, and V a verbal subgroup belonging to G′. Following H. Neumann [6] we shall use the vector representation for endomorphisms of G. Vector v = (ν1, ν2,…) represents an endomorphism v such that xiv = νi for all i. The identity map is represented by l=(x1,x2…). We need also thetrivial endomorphism 0 = (e, e,…). The length of vectors is equal to the rank of G. We shall consider the near-ring of vectors, with addition and multiplication given below u + v=(ulν1, u2ν2,…) where uiνi; is a product in G, and uv = (u1v, u2v,…) where uiv isthe image of ui, under the endomorphism v. There is only one distributivity law (u + v)w =uw + vw.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 1 , February 1987 , pp. 115 - 120
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Bachmuth, S., Induced automorphisms of free groups and free metabelian groups, Trans. Amer. Math. Soc. 122 (1966), 1–17.Google Scholar
2.Hall, P., Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2 (1958), 787–801.CrossRefGoogle Scholar
3.Kaloujnine, L., Sur quelques proprietes des groupes d'automorphismes d'un groupe abstrait, C.R. Acad. Sci. Paris 231 (1950), 400–402.Google Scholar
4.Malcev, A., On algebras with identical relations, Matem. Sbornik 26 (1) (1950), 19–33.Google Scholar
5.Mostowski, W., On automorphisms of relatively free groups, Fund. Math. 50 (1961/1962), 403–411.Google Scholar
6.Neumann, H., On varieties of groups and their associated near-rings, Math. Z. 65 (1956), 36–69.CrossRefGoogle Scholar
7.Neumann, H., Varieties of Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 37, Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
8.Plotkin, B., On a radical of an automorphism group of a group with maximal condition, Dokl. Akad. Nauk SSSR 130 (1960), 977–980.Google Scholar
9.Viljcer, V., Stable groups of automorphisms, Dokl. Akad. Nauk SSSR 131 (1960), 728–730.Google Scholar