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A tag, a kind, a name, an inventory row, and a same_as or a source.

Ideas 451–600 of 6044.

Name	Kind	Subject
Product of cyclic groups is cyclic iff the orders are coprime	Theorem	Group theory and generalizations
Group of units of a monoid	Definition	Group theory and generalizations
Homomorphisms preserve identity and inverses	Proposition	Group theory and generalizations
Product of countable sets is countable	Proposition	Mathematical logic and foundations
Uniform convergence on a set	Definition	Sequences, series, summability
Fixed points of an action	Definition	Group theory and generalizations
Group isomorphism	Concept	Group theory and generalizations
Distributive laws for sets	Theorem	Mathematical logic and foundations
Infinite limit	Definition	Real functions
Lower central series	Definition	Group theory and generalizations
Product formula for the order of HK	Proposition	Group theory and generalizations
Picard's great theorem	Theorem	Functions of a complex variable
The sequence 1/n tends to zero	Example	Sequences, series, summability
Independence means the joint law is the product of the marginals	Proposition	Probability theory and stochastic processes
Number of functions between finite sets	Theorem	Combinatorics
Number of distinct prime factors	Concept	Number theory
Power series solution	Concept	Ordinary differential equations
Differentiable on an open set	Definition	Real functions
Topological embedding	Definition	General topology
Order of an element divides the order of the group	Corollary	Group theory and generalizations
Semisimple endomorphism	Concept	Linear and multilinear algebra; matrix theory
Content of a polynomial	Definition	Commutative algebra
Power series of cosine	Theorem	Special functions
Hall's marriage theorem	Theorem	Combinatorics
Sine and cosine	Concept	Special functions
Euler's four-square identity	Proposition	Number theory
Explicit and implicit differential equations	Concept	Ordinary differential equations
An iterate of the Picard operator is a contraction	Lemma	Ordinary differential equations
Comparison of topologies	Definition	General topology
Subfields of a finite field	Theorem	Field theory and polynomials
Identically distributed random variables	Concept	Probability theory and stochastic processes
Eigenvalue method for linear systems	Concept	Ordinary differential equations
One-sided limit	Definition	Real functions
Convergence of a sequence in a metric space	Concept	General topology
An injective endomorphism of a finite-dimensional space is surjective	Theorem	Linear and multilinear algebra; matrix theory
Real exponentiation	Concept	Real functions
Geometric probability	Concept	Probability theory and stochastic processes
Congruent triangles	Concept	Geometry
Mesh of a partition	Concept	Real functions
Minimal polynomial divides the characteristic polynomial	Corollary	Linear and multilinear algebra; matrix theory
Multiplication by a matrix as a linear map	Concept	Linear and multilinear algebra; matrix theory
Row equivalence	Definition	Linear and multilinear algebra; matrix theory
In a PID a nonzero element is irreducible iff it generates a maximal ideal	Theorem	Commutative algebra
Extended Euclidean algorithm	Concept	Number theory
Young diagram	Concept	Combinatorics
Unique paths in trees	Theorem	Combinatorics
Subnormal series	Definition	Group theory and generalizations
Geometric solutions come from roots of the characteristic polynomial	Theorem	Difference and functional equations
Linearly independent set of the right size is a basis	Proposition	Linear and multilinear algebra; matrix theory
Riesz's lemma	Lemma	Functional analysis
Domain in the complex plane	Definition	Functions of a complex variable
Intermediate value theorem for connected spaces	Theorem	General topology
Fixed points of a p-group action	Lemma	Group theory and generalizations
Gauss–Jordan elimination	Concept	Linear and multilinear algebra; matrix theory
Space of solutions of a linear recurrence	Concept	Difference and functional equations
Geometric multiplicity	Concept	Linear and multilinear algebra; matrix theory
Sequence tending to infinity	Definition	Sequences, series, summability
Power series of the complex logarithm	Proposition	Functions of a complex variable
Necessary and sufficient conditions	Concept	Mathematical logic and foundations
Area of a region as a line integral	Corollary	Real functions
Open sets characterised by neighbourhoods	Proposition	General topology
Coordinates in an orthonormal basis	Proposition	Linear and multilinear algebra; matrix theory
Trace of a transpose	Proposition	Linear and multilinear algebra; matrix theory
Complex differentiability	Concept	Functions of a complex variable
Circuit	Concept	Combinatorics
Upper and lower Darboux integrals	Definition	Real functions
Epsilon-N definition of the limit of a sequence	Proposition	Sequences, series, summability
Second Sylow theorem	Theorem	Group theory and generalizations
Transpose of a product	Proposition	Linear and multilinear algebra; matrix theory
Closed sets contain their limits	Proposition	General topology
Completeness of the space of bounded continuous functions	Theorem	General topology
Method of moments	Concept	Statistics
Forest	Definition	Combinatorics
Sum of the totient over the divisors	Theorem	Number theory
Groups of order p squared are abelian	Theorem	Group theory and generalizations
Order of a differential equation	Definition	Ordinary differential equations
Ring homomorphisms from a field are injective	Proposition	Field theory and polynomials
Finite intersection of open sets is open	Proposition	General topology
Mean of the normal distribution	Proposition	Probability theory and stochastic processes
Liouville's formula	Theorem	Ordinary differential equations
Cancellation in congruences	Proposition	Number theory
The rationals form an ordered field	Theorem	Field theory and polynomials
Smooth non-analytic function	Counterexample	Real functions
Equivalent metrics	Concept	General topology
Scalar multiplication of matrices	Concept	Linear and multilinear algebra; matrix theory
Orthonormal family	Definition	Linear and multilinear algebra; matrix theory
Every walk contains a path	Proposition	Combinatorics
Permuting rows multiplies the determinant by the sign	Proposition	Linear and multilinear algebra; matrix theory
Divisibility by 11	Proposition	Number theory
Intersection of closed sets is closed	Proposition	General topology
The number of Sylow subgroups is the index of the normaliser	Proposition	Group theory and generalizations
Product of compact sets is compact	Theorem	General topology
Linearisation at an equilibrium	Concept	Ordinary differential equations
Higher partial derivatives	Concept	Real functions
Determinant of a matrix exponential	Theorem	Linear and multilinear algebra; matrix theory
Open sets of a metric space	Proposition	General topology
Uniform continuity on a set	Definition	Real functions
The reals are Archimedean	Theorem	Real functions
Linear equation with constant coefficients	Definition	Ordinary differential equations
Derivative gives the directional derivatives	Proposition	Real functions
System of linear differential equations	Definition	Ordinary differential equations
Metric topology	Definition	General topology
Archimedean property	Proposition	Field theory and polynomials
Tower law	Theorem	Field theory and polynomials
Complex sine and cosine	Definition	Special functions
Bounded sequence	Concept	Sequences, series, summability
x^2 is not uniformly continuous on the real line	Counterexample	Real functions
Chain rule for a composition with a curve	Corollary	Real functions
Complete ordered field	Concept	Field theory and polynomials
Periodic orbit	Concept	Dynamical systems and ergodic theory
Kernel of a ring homomorphism	Definition	Associative rings and algebras
Least element	Concept	Order, lattices, ordered algebraic structures
Almost sure convergence	Concept	Probability theory and stochastic processes
Finite-dimensional normed spaces are complete	Theorem	Functional analysis
Metrizable spaces are normal	Proposition	General topology
Formal derivative	Definition	Commutative algebra
Strictly convex function	Concept	Real functions
sin(1/x) has no limit at 0	Counterexample	Real functions
If G modulo its centre is cyclic then G is abelian	Proposition	Group theory and generalizations
Generalised eigenspace	Concept	Linear and multilinear algebra; matrix theory
Multiplication by a matrix as a linear map	Definition	Linear and multilinear algebra; matrix theory
Independent and identically distributed random variables	Definition	Probability theory and stochastic processes
Successor function	Concept	Mathematical logic and foundations
Classical definition of probability	Definition	Probability theory and stochastic processes
Gram matrices are positive semidefinite	Proposition	Linear and multilinear algebra; matrix theory
Cauchy product of absolutely convergent series	Theorem	Sequences, series, summability
Circle integral	Concept	Functions of a complex variable
Nullity	Concept	Linear and multilinear algebra; matrix theory
Second isomorphism theorem for vector spaces	Theorem	Linear and multilinear algebra; matrix theory
The symmetric group is generated by transpositions	Proposition	Group theory and generalizations
Abelianisation	Concept	Group theory and generalizations
Pairwise independent events that are not mutually independent	Counterexample	Probability theory and stochastic processes
Simplicity of the alternating group	Theorem	Group theory and generalizations
Power sum symmetric polynomials	Definition	Combinatorics
Consistent estimator	Concept	Statistics
The Möbius function is the inverse of the constant function 1	Theorem	Number theory
Annihilator of a subspace	Concept	Linear and multilinear algebra; matrix theory
Normaliser	Definition	Group theory and generalizations
Back substitution	Concept	Linear and multilinear algebra; matrix theory
Boundedness theorem	Theorem	Real functions
The nth root of n tends to 1	Example	Sequences, series, summability
R/P is an integral domain iff P is prime	Theorem	Commutative algebra
Totient of a prime power	Proposition	Number theory
K[x]/(f) is a field iff f is irreducible	Theorem	Field theory and polynomials
Marginal distribution	Definition	Statistics
Completeness axiom	Axiom	Real functions
Eigenspace	Definition	Linear and multilinear algebra; matrix theory
Stabiliser	Concept	Group theory and generalizations
Product metric	Concept	General topology
Dual map	Concept	Linear and multilinear algebra; matrix theory