For finite sets X, the axiom of choice follows from the other axioms of set theory.

For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

For a finite group, the derived series terminates in a perfect group, which may or may not be trivial.

For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.

* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.

For example, Graham's number, though finite, is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes ' number and Moser's number.

For some finite n-valued logics, there is an analogous law called the law of excluded n + 1th.

For every finite dimensional matrix Lie algebra, there is a linear group ( matrix Lie group ) with this algebra as its Lie algebra.

Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values.

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite.

For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits.

For a nonzero vector of finite norm in, one can assume that is nonzero, or to fix ideas.

For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.

For any luminosity from a given distance L ( r ) N ( r ) proportional to r < sup > a </ sup >, is infinite for a ≥ − 1 but finite for a < − 1.

For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative ( since it decreases from zero ) for smaller finite distances.

For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions.

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.