Examples of "nilpotent"
In mathematics, in the field of group theory, a metanilpotent group is a group that is nilpotent by nilpotent. In other words, it has a normal nilpotent subgroup such that the quotient group is also nilpotent.
A nilpotent element is an element formula_26 such that formula_31 for some formula_32. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.
In , it was proven that a ring which is the direct sum of two nilpotent subrings is itself nilpotent. The question arose whether or not "nilpotent" could be replaced with "locally nilpotent" or "nil". Partial progress was made when Kelarev produced an example of a ring which isn't nil, but is the direct sum of two locally nilpotent rings. This demonstrates that Kegel's question with "locally nilpotent" replacing "nilpotent" is answered in the negative.
This matrix is nilpotent with degree "n", and is the “canonical” nilpotent matrix.
A locally nilpotent ring is one in which every finitely generated subring is nilpotent: locally nilpotent rings form a radical class, giving rise to the Levitzki radical.
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.
However, nilpotent normal subgroups do not in general form a complete lattice, as a subgroup generated by an infinite collection of nilpotent normal subgroups need not be nilpotent, though it will be normal. The join of all nilpotent normal subgroups is still defined as the Fitting subgroup, but it need not be nilpotent.
This is not a defining characteristic of nilpotent groups: groups for which formula_18 is nilpotent of degree "n" (in the sense above) are called "n"-Engel groups, and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated.
In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role
Let "Q" be a loop whose inner mapping group is nilpotent. Is "Q" nilpotent? Is "Q" solvable?
If "G" ≠ "G" for all finite "n", then "G"/"G" is not nilpotent, but it is residually nilpotent.
Theorem. A finite-dimensional Lie algebra L is nilpotent if and only if every element of L is ad-nilpotent.
Let "R" be a right Noetherian ring. Then every nil one-sided ideal of "R" is nilpotent. In this case, the upper and lower nilradicals are equal, and moreover this ideal is the largest nilpotent ideal among nilpotent right ideals and among nilpotent left ideals.
Every subgroup of a nilpotent group of class "n" is nilpotent of class at most "n"; in addition, if "f" is a homomorphism of a nilpotent group of class "n", then the image of "f" is nilpotent of class at most "n".
is nilpotent. Note that in the Lie algebra "L"("V") of linear operators on "V", the identity operator I is ad-nilpotent (because formula_3) but is not a nilpotent operator.
Every [[nilpotent group]] or Lie algebra is Engel. [[Engel's theorem]] states that every finite-dimensional Engel algebra is nilpotent. gave examples of a non-nilpotent Engel groups and algebras.
A Carter subgroup is a maximal nilpotent subgroup, because of the normalizer condition for nilpotent groups, but not all maximal nilpotent subgroups are Carter subgroups . For example, any non-identity proper subgroup of the nonabelian group of order six is a maximal nilpotent subgroup, but only those of order two are Carter subgroups. Every subgroup containing a Carter subgroup of a soluble group is also self-normalizing, and a soluble group is generated by any Carter subgroup and its nilpotent residual .
A dual notion to Carter subgroups was introduced by Bernd Fischer in . A Fischer subgroup of a group is a nilpotent subgroup containing every other nilpotent subgroup it normalizes. A Fischer subgroup is a maximal nilpotent subgroup, but not every maximal nilpotent subgroup is a Fischer subgroup: again the nonabelian group of order six provides an example as every non-identity proper subgroup is a maximal nilpotent subgroup, but only the subgroup of order three is a Fischer subgroup .
Every nilpotent group is "p"-nilpotent, and every "p"-nilpotent group is "p"-soluble. Every soluble group is "p"-soluble, and every "p"-soluble group is "p"-constrained. A group is "p"-nilpotent if and only if it has a normal "p"-complement, which is just its "p"′-core.
In 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. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.