mathematics 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 ...

, especially in the study of infinite groups, the Hirsch–Plotkin radical is a

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

describing the normal locally nilpotent subgroups of the group. It was named by after

Kurt Hirsch Kurt August Hirsch (12 January 1906 – 4 November 1986) was a Germans, German mathematician who moved to England to escape the Nazi persecution of Jews. His research was in group theory. He also worked to reform mathematics education and became ...

and Boris I. Plotkin, who proved that the join of normal locally nilpotent subgroups is locally nilpotent; this fact is the key ingredient in its construction. The Hirsch–Plotkin radical is defined as the subgroup generated by the union of the normal locally nilpotent subgroups (that is, those normal subgroups such that every finitely generated subgroup is nilpotent). The Hirsch–Plotkin radical is itself a locally nilpotent normal subgroup, so is the unique largest such. In a finite group, the Hirsch–Plotkin radical coincides with 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 ...

but for infinite groups the two subgroups can differ. The subgroup generated by the union of infinitely many normal nilpotent subgroups need not itself be nilpotent, so the Fitting subgroup must be modified in this case.. Se
p. 40
"In general the Fitting subgroup in an infinite group gives little information about the structure of the group".

References

Functional subgroups Infinite group theory {{algebra-stub