For purposes of comparison, consider the nilradical of a commutative ring, which consists of all elements which are nilpotent.

For example, the ring Zy /( xy ) contains x + ( xy ) and y + ( xy ) as zero divisors, but no non-zero nilpotent elements.

For algebraically closed fields of characteristic p > 0 Lie's theorem holds provided the dimension of the representation is less than p, but can fail for representations of dimension p. An example is given by the 3-dimensional nilpotent Lie algebra spanned by 1, x, and d / dx acting on the p-dimensional vector space k /( x < sup > p </ sup >), which has no eigenvectors.

* The degree of an n × n nilpotent matrix is always less than or equal to n. For example, every 2 × 2 nilpotent matrix squares to zero.

For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix

* A module is called coprimary if every zero divisor of M is nilpotent in M. For example, groups of prime power order and free abelian groups are coprimary modules over the ring of integers.

For example, any operator given by a matrix whose entries are zero on and below its diagonal is nilpotent.

For an arbitrary ( not necessarily finite ) group G, the Fitting subgroup is defined to be the subgroup generated by the nilpotent normal subgroups of G. For infinite groups, the Fitting subgroup is not always nilpotent.

For it includes the emotional ties that bind men to their homeland and the complex motivations that hold a large group of people together as a unit.

For ten years a small group of European and U.S. critics has been calling attention to the half-forgotten Austrian expressionist Egon Schiele, who died 42 years ago at the age of 28.

For example, the steering committee might announce that the group felt a topic under study should not be dropped for an additional week as there was still too much of it untouched.

For purposes of sample selection only ( individual tests were given later ) we obtained group test scores of reading achievement and intelligence from school records of the entire third-grade population in each school system.

For if the small group notion involves the implicit claim that the phenomena of sociological investigations are of atomic or subatomic proportions, the philosopher needs to know the extent to which such entities are valid.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

For every group G there is a natural group homomorphism G → Aut ( G ) whose image is the group Inn ( G ) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can be embedded into its own automorphism group.

For each element a of a group G, conjugation by a is the operation φ < sub > a </ sub >: G → G given by ( or a < sup >− 1 </ sup > ga ; usage varies ).

For example, three isomers exist for cresol because the methyl group and the hydroxyl group can be placed next to each other ( ortho ), one position removed from each other ( meta ), or two positions removed from each other ( para ).

For the bottom, only the group of higher priority need be considered.

For the U. S. Government's scientific enterprise, a significant impact of NAPAP were lessons learned in the assessment process and in environmental research management to a relatively large group of scientists, program managers and the public.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

For a few years, this group of animals was regarded as a subfamily, called the Callitrichinae, of the family Cebidae.

For example, it is thought that ammonites were the principal food of mosasaurs, a group of giant marine reptiles that became extinct at the boundary .< ref name =" Kauffman ">

For example, an " X " is used to indicate a variable group amongst a class of compounds ( though usually a halogen ), while " R " is used for a radical, meaning a compound structure such as a hydrocarbon chain.

For example, it is thought that ammonites were the principal food of mosasaurs, a group of giant marine reptiles that became extinct at the boundary .< ref name =" Kauffman ">

For example, it is immediately proven from the axioms that the identity element of a group is unique.

For four of the six ( the anaconda and the amethystine python cannot be included for lack of data ) there is also a correlation between size at maturity and maximum length, the boa constrictor being the smallest and the Indian python the next in size at the former stage.

For example, the smallest pair of amicable numbers is ( 220, 284 ); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284 ; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.

For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is defined to be the smallest such distance.

* K < nowiki ></ nowiki > X < nowiki ></ nowiki >, the ring of formal power series over the field K. For each nonzero power series P, define f ( P ) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f ( P )≤ f ( Q ) iff P divides Q.

For any subset N of M, the monoid N * is the smallest monoid that contains N.

For example, the prime field is the smallest subfield of a field F containing both 0 and 1.

For a simple example of the first kind, consider the proposition, ¬ p: " there is no smallest rational number greater than 0 ".

For most certainly, I tell you, until heaven and earth pass away, not even one smallest letter or one tiny pen stroke shall in any way pass away from the law, until all things are accomplished.

For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample ( i. e., a natural number n for which the Mertens function M ( n ) equals or exceeds the square root of n ) is known: all numbers less than 10 < sup > 14 </ sup > have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1. 59 × 10 < sup > 40 </ sup >, which is approximately 10 to the power 4. 3 × 10 < sup > 39 </ sup >.

For several years after his marriage Morris was absorbed in two connected occupations: the building and decoration of a house for himself and Jane, and the foundation of a firm of decorators who were also artists, with the view of reinstating decoration, down to its smallest details, as one of the fine arts.

For this class of profiles, multimode distortion is smallest when takes a particular value depending on the material used.

For example, if you started with three pieces, you would move the smallest piece to the opposite end, then continue in the left direction after that.

For the very first move, the smallest disk goes to peg t if h is odd and to peg r if h is even.

For the smallest disk there are always two possibilities.

For the other disks there is always one possibility, except when all disks are on the same peg, but in that case either it is the smallest disk that must be moved or the objective has already been achieved.

For example, unlike the other alkali metals the smallest one, lithium, does not form alums, and there is only one known sodium alum.

* For a given formal language, what is the smallest automaton that recognizes it?

For years, it has been the smallest populated town in Alabama, but the town is notable for the nearby rock arch, also named Natural Bridge, the longest natural bridge east of the Rockies, spanning over 127 ft ( 40 m ).

For the most recent data available, it tied with Sweetwater for the smallest high school with 15 students.

* For a small non-abelian example, consider the quaternion group Q < sub > 8 </ sub >, which is a smallest non-abelian p-group.

For example, American Mensa has 134 local groups, with the largest having over 2, 000 members and the smallest having fewer than 100.

For example, for the smallest known Sierpinski number, 78557, the covering set is < nowiki >

For any T, a smallest T-join necessarily consists of | T | paths, no two having an edge in common, that join the vertices of T in pairs.

For example, there is a difference of as much as 0. 045 inches ( 1. 15 mm ) between the smallest and largest of the several cartridges designated as '. 38 caliber '.