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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

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