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On Special Group-Automorphisms and Their Composition

Published online by Cambridge University Press:  20 November 2018

Peter Hilton
Peter Hilton*
Affiliation:
Battelle Human Affairs Research Centres, Seattle, Washington
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Let G be a group and φ an automorphism of G. We say that φ is a pseudo-identity (pi) if, for each x ∈ G, there exists a finitely generated (fg) subgroup K = Kx(φ) of G such that x ∈ K and φ|K is an automorphism of K. It has been shown [1, 3] that such special automorphisms of abelian or nilpotent groups play an important role in homotopy theory; and it was indicated in [2] that their purely algebraic properties might well repay study.

The following facts about pi's are elementary.

PROPOSITION 0.1. Let φ be an automorphism of G and let n be a non-zerointeger. Then φ is pi if and only if φn is pi.

PROPOSITION 0.2. Let φ be a pseudo-identity of G and a an automorphismof G. Then αφα-1 is a pseudo-identity.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 4 , 01 August 1984 , pp. 591 - 600
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Cohen, J. M., A spectral sequence automorphism theorem: applications to Jihre spaces and stable homotopy, Topology 7 (1968), 173–177.Google Scholar
2. Hilton, P. and Roitberg, J., On pseudo-identities, I Archiv der Mathematik 41 (1983), 204–214.Google Scholar
3. Hilton, P., Castellet, M. and Roitberg, J., On pseudo-identities II, Archiv der Mathematik 42 (1984), 193–199.Google Scholar
4. Hilton, P., Mislin, G. and Roitberg, J., Localization of nilpotent groups and spaces, Mathematics Studies 15 (North Holland, 1975).Google Scholar
5. Hilton, P., On direct limits of nilpotent groups, Springer Lecture Notes 418 (1974), 68–77.Google Scholar
6. Stammbach, U., Homology in group theory, Springer Lecture Notes 359 (1973).CrossRefGoogle Scholar