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“Hasse principle” for \(\text{GL}_n(D)\). (English) Zbl 0967.20027

Let \(D\) be a Euclidean ring, \(n\in\mathbb{N}\) and \(\text{GL}_n(D)\) the \(n\)-th general linear group over \(D\). If \(f\) is an endomorphism of \(\text{GL}_n(D)\) which preserves conjugacy classes then \(f\) is already an inner automorphism.

MSC:

20G35 Linear algebraic groups over adèles and other rings and schemes
20E36 Automorphisms of infinite groups
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
Full Text: DOI

References:

[1] Hua, L. K.: Introduction to Number Theory. Springer-Verlag, New York, pp. 371-382 (1982).
[2] Ono, T.: “Hasse principle” for \(GL_2(D)\). Proc. Japan Acad., 75A , 141-142 (1999). · Zbl 0968.20024 · doi:10.3792/pjaa.75.141
[3] Wada, H.: “Hasse principle” for \(SL_n(D)\). Proc. Japan Acad., 75A , 67-69 (1999). · Zbl 1041.11025 · doi:10.3792/pjaa.75.67
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