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Complete Reductibility of Infinite Groups

Published online by Cambridge University Press:  20 November 2018

John D. Dixon
John D. Dixon*
Affiliation:
California Institute of Technology
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The theorems of the present paper deal with conditions which are necessary and sufficient in order that a solvable or nilpotent infinite group should have a completely reducible matrix representation over a given algebraically closed field.

It is known (17) that a locally nilpotent group of matrices is always solvable. Thus the first theorem of the present paper is a partial generalization of Theorem 1 of (16), which states:

If G is a locally nilpotent subgroup of the full linear group GL(n, P) over a perfect field P, then G is completely reducible if and only if each matrix of G is diagonizable (by a similarity transformation over some extension field of P).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 267 - 274
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Baer, R., Finiteness properties of groups, Duke Math. J., 15 (1948), 1021–1032.Google Scholar
2. Burnside, W., On criteria for the finiteness of the order of a group of linear substitutions, Proc. Lond. Math. Soc. (2), 3 (1905), 435–440.Google Scholar
3. Clifford, A. H., Representations induced in an invariant subgroup, Ann. Math., 38 (1937), 533–550.Google Scholar
4. Fuchs, L., Abelian groups (Budapest, 1958).Google Scholar
5. Hall, M., Theory of groups (New York, 1959).Google Scholar
6. Hall, P., Nilpotent groups, Can. Math. Congress, Summer Seminar (1957).Google Scholar
7. Huppert, B., Lineare auflôsbare Gruppen, Math. Z., 67 (1957), 479–518.Google Scholar
8. Kurosh, A. G., Theory of groups (New York, 1955).Google Scholar
9. Mal, A. I.'cev, On some classes of infinite solvable groups, Mat. Sborn, N.S. #5(70) (1951), 567-588 [Am. Math. Soc. Translations (2), 2 (1956), 1–22].Google Scholar
10. Neumann, B. H., Groups with finite classes of conjugate elements, Proc. Lond. Math. Soc. (3), 1 (1951), 178–187.Google Scholar
11. Suprunenko, D. A., Solvable groups of matrices, Doklady Akad. Nauk, SSSR, 83 (1952), 183–186 (Russian).Google Scholar
12. Neumann, B. H., Locally nilpotent irreducible subgroups of the full linear group, Doklady Akad. Nauk SSSR, 102 (1955), 41–44 (Russian).Google Scholar
13. Neumann, B. H., On linear solvable groups, Mat. Sborn., N.S. 41 (82) (1957), 317–322 (Russian).Google Scholar
14. Neumann, B. H., Solvable and nilpotent linear groups (Minsk, 1958) (Russian). [Transi. Math. Monographs, vol. 9 (Providence, 1963).]Google Scholar
15. Neumann, B. H., The condition of complete reducibility of a solvable linear group, Doklady Akad. Nauk BSSR, 5 (1961), 321–323 (Russian) (see Math. Rev., 25 (1963), No. 125).Google Scholar
16. Suprunenko, D. A. and Tyshkevich, R., Reducible locally nilpotent linear groups, Izv. Akad. Nauk SSSR, 24 (1960), 787–806 (Russian).Google Scholar
17. Zassenhaus, H., Beweis eines Satzes über diskrete Gruppen, Abh. Math. Sem. Hamburg, 12 (1938), 289–312.Google Scholar