Geodesic
On Special Group-Automorphisms and Their Composition
Canadian journal of mathematics, Tome 36 (1984) no. 4, pp. 591-600

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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 = K x(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. 
Hilton, Peter. On Special Group-Automorphisms and Their Composition. Canadian journal of mathematics, Tome 36 (1984) no. 4, pp. 591-600. doi: 10.4153/CJM-1984-037-6
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[1] 1. Cohen, J. M., A spectral sequence automorphism theorem: applications to Jihre spaces and stable homotopy, Topology 7 (1968), 173–177. Google Scholar

[2] 2. Hilton, P. and Roitberg, J., On pseudo-identities, I Archiv der Mathematik 41 (1983), 204–214. Google Scholar

[3] 3. Hilton, P., Castellet, M. and Roitberg, J., On pseudo-identities II, Archiv der Mathematik 42 (1984), 193–199. Google Scholar

[4] 4. Hilton, P., Mislin, G. and Roitberg, J., Localization of nilpotent groups and spaces, Mathematics Studies 15 (North Holland, 1975). Google Scholar

[5] 5. Hilton, P., On direct limits of nilpotent groups, Springer Lecture Notes 418 (1974), 68–77. Google Scholar

[6] 6. Stammbach, U., Homology in group theory, Springer Lecture Notes 359 (1973). Google Scholar | DOI

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