Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T01:17:37.708Z Has data issue: false hasContentIssue false
  • English
  • Français

The finite inner automorphism groups of division rings

Published online by Cambridge University Press:  24 October 2008

M. Shirvani
M. Shirvani
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1 e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 2 , September 1995 , pp. 207 - 213
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Artin, E.. Geometrie Algebra (Wiley, 1988).Google Scholar
[2]Cohn, P. M.. Skew Field Constructions (Cambridge University Press, 1977).Google Scholar
[3]Faudree, R. J.. Subgroups of the multiplicative group of a division ring. Trans. Amer. Math. Soc. 124 (1966), 41–48.Google Scholar
[4]Huppert, B.. Endliche Gruppen I (Springer-Verlag, 1967).CrossRefGoogle Scholar
[5]Jones, M. R.. Some inequalities for the multiplicator of a finite group. Proc. Amer. Math. Soc. 45 (1974), 167–172.Google Scholar
[6]Karpilovsky, G.. The Schur multiplier (Oxford, 1987).Google Scholar
[7]Montgomery, S.. Fixed rings of finite automorphism groups of associative rings (Springer-Verlag, 1980).Google Scholar
[8]Passman, D. S.. Permutation groups (W. A. Benjamin, 1968).Google Scholar
[9]Shirvani, M.. The structure of simple rings generated by finite metabelian groups. J. Alg., 169 (1994), 686–712.CrossRefGoogle Scholar
[10]Shirvani, M. and Wehrfritz, B. A. F.. Skew linear groups (Cambridge University Press, 1987).Google Scholar