Fitting's theorem
Fitting's theorem is a mathematical theorem proved by Hans Fitting.[1] It can be stated as follows:
- If M and N are nilpotent normal subgroups of a group G, then their product MN is also a nilpotent normal subgroup of G; if, moreover, M is nilpotent of class m and N is nilpotent of class n, then MN is nilpotent of class at most m + n.[2]
By induction it follows also that the subgroup generated by a finite collection of nilpotent normal subgroups is nilpotent. This can be used to show that the Fitting subgroup of certain types of groups (including all finite groups) is nilpotent. However, a subgroup generated by an infinite collection of nilpotent normal subgroups need not be nilpotent.[3]
References
[edit]- ^ Fitting, Hans (1938), "Beiträge zur Theorie der Gruppen endlicher Ordnung", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 48: 77–141; see Hilfsatz 10 (unnumbered in text), p. 100
- ^ Clement, Anthony E.; Majewicz, Stephen; Zyman, Marcos (2017), "2.3.6 Products of Normal Nilpotent Subgroups", The theory of nilpotent groups, Cham: Birkhäuser/Springer, pp. 46–47, doi:10.1007/978-3-319-66213-8, ISBN 978-3-319-66211-4
- ^ Clement, Majewicz & Zyman (2017), Lemma 7.18 and Remark 7.8, p. 297
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