Fitting's theorem

Fitting's theorem is a mathematical theorem proved by Hans Fitting.{{citation

| last = Fitting | first = Hans | author-link = Hans Fitting

| journal = Jahresbericht der Deutschen Mathematiker-Vereinigung

| language = de

| pages = 77–141

| title = Beiträge zur Theorie der Gruppen endlicher Ordnung

| url = https://eudml.org/doc/146181

| volume = 48

| year = 1938}}; 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.{{citation

| last1 = Clement | first1 = Anthony E.

| last2 = Majewicz | first2 = Stephen

| last3 = Zyman | first3 = Marcos

| contribution = 2.3.6 Products of Normal Nilpotent Subgroups

| doi = 10.1007/978-3-319-66213-8

| isbn = 978-3-319-66211-4

| pages = 46–47

| publisher = Birkhäuser/Springer | location = Cham

| title = The theory of nilpotent groups

| year = 2017}}

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.{{harvtxt|Clement|Majewicz|Zyman|2017}}, Lemma 7.18 and Remark 7.8, p. 297