Hirsch–Plotkin radical
Encyclopedia
In mathematics
, especially in the study of infinite groups
, the Hirsch–Plotkin radical is a subgroup describing the normal
nilpotent
subgroup
s of the group.
The Hirsch–Plotkin radical is defined as the subgroup generated by 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. The Hirsch–Plotkin radical generalizes the Fitting subgroup
to infinite groups. Unfortunately the subgroup generated by infinitely many normal nilpotent subgroups need not itself be nilpotent, so the Fitting subgroup must be modified in this case.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the study of infinite groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, the Hirsch–Plotkin radical is a subgroup describing the normal
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
nilpotent
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
s of the group.
The Hirsch–Plotkin radical is defined as the subgroup generated by 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. The Hirsch–Plotkin radical generalizes the Fitting subgroup
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 smallest subgroup which "controls" the structure of G when G is solvable...
to infinite groups. Unfortunately the subgroup generated by infinitely many normal nilpotent subgroups need not itself be nilpotent, so the Fitting subgroup must be modified in this case.