Fitting's theorem
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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 | edit source]- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).; see Hilfsatz 10 (unnumbered in text), p. 100
- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- ^ Clement, Majewicz & Zyman (2017), Lemma 7.18 and Remark 7.8, p. 297
