Fitting's theorem

Fitting's theorem is a mathematical theorem proved by Hans Fitting.; see Hilfsatz 10 (unnumbered in text), p. 100 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.

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., Lemma 7.18 and Remark 7.8, p. 297

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