Fitting's theorem is a

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

proved by Hans Fitting. It can be stated as follows: :If ''M'' and ''N'' are

nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

s of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

''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 In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup ''F'' of a finite group ''G'', named after Hans Fitting, is the unique largest normal nilpotent subgroup of ''G''. Intuitively, it represents the smalle ...

of certain types of groups (including all

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

s) 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

References

Theorems in group theory {{Abstract-algebra-stub