nilpotentadj.

Brit.

/nɪlˈpəʊtnt/, U.S.

/ˌnɪlˈpoʊtnt/,

/ˌnɪlˈpoʊdənt/

Origin: Formed within English, by compounding. Etymons: nil n.¹, potent adj.¹

Etymology: < nil n.¹ + potent adj.¹

Mathematics.

Of a function, matrix, group element, etc.: becoming zero when raised to some positive integral power. Of a group: becoming trivial, containing only the identity element, when raised to some positive integral power.The operation of raising a group to a power involves successive applications of the commutator operator: G⁰ ≡ G, and Gⁱ ≡ [Gⁱ⁻¹, G], the set of all elements of the form h⁻¹g⁻¹hg, where h ∈ Gⁱ⁻¹, g ∈ G.

ΘΚΠ

the world > relative properties > number > mathematical number or quantity > [adjective] > zero > becoming zero

vanishing1823

nilpotent1870

1870 B. Peirce in Amer. Jrnl. Math. (1881) 4 104 When an expression raised to the square or any higher power vanishes, it may be called nilpotent.

1937 A. A. Albert Mod. Higher Algebra (1938) iv. 87 Two nilpotent matrices of index two are similar in F if and only if they have the same rank.

1949 S. Kravetz tr. H. Zassenhaus Theory of Groups iv. 111 A group G is said to be nilpotent if the ascending central series contains the whole group as a member.

1971 I. T. Adamson Rings, Modules & Algebras xxiii. 194 We say that I is a nilpotent ideal if there exists a positive integer n such that Iⁿ is the zero ideal.

1990 Q. Jrnl. Math. 41 127 Let A be a finite group, and suppose there are distinct primes p and q such that A is neither p-nilpotent nor q-nilpotent. Then there are finite {p, q}-groups of arbitrarily large Fitting height, on which A operates fixed-point-freely.

This entry has been updated (OED Third Edition, September 2003; most recently modified version published online March 2022).

单词	nilpotent
释义	nilpotentadj. Brit. /nɪlˈpəʊtnt/, U.S. /ˌnɪlˈpoʊtnt/, /ˌnɪlˈpoʊdənt/ Origin: Formed within English, by compounding. Etymons: nil n.¹, potent adj.¹ Etymology: < nil n.¹ + potent adj.¹ Mathematics. Of a function, matrix, group element, etc.: becoming zero when raised to some positive integral power. Of a group: becoming trivial, containing only the identity element, when raised to some positive integral power.The operation of raising a group to a power involves successive applications of the commutator operator: G⁰ ≡ G, and Gⁱ ≡ [Gⁱ⁻¹, G], the set of all elements of the form h⁻¹g⁻¹hg, where h ∈ Gⁱ⁻¹, g ∈ G. ΘΚΠ the world > relative properties > number > mathematical number or quantity > [adjective] > zero > becoming zero vanishing1823 nilpotent1870 1870 B. Peirce in Amer. Jrnl. Math. (1881) 4 104 When an expression raised to the square or any higher power vanishes, it may be called nilpotent. 1937 A. A. Albert Mod. Higher Algebra (1938) iv. 87 Two nilpotent matrices of index two are similar in F if and only if they have the same rank. 1949 S. Kravetz tr. H. Zassenhaus Theory of Groups iv. 111 A group G is said to be nilpotent if the ascending central series contains the whole group as a member. 1971 I. T. Adamson Rings, Modules & Algebras xxiii. 194 We say that I is a nilpotent ideal if there exists a positive integer n such that Iⁿ is the zero ideal. 1990 Q. Jrnl. Math. 41 127 Let A be a finite group, and suppose there are distinct primes p and q such that A is neither p-nilpotent nor q-nilpotent. Then there are finite {p, q}-groups of arbitrarily large Fitting height, on which A operates fixed-point-freely. This entry has been updated (OED Third Edition, September 2003; most recently modified version published online March 2022). < adj.1870
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