The inner automorphisms form a normal subgroup of Aut ( G ), denoted by Inn ( G ); this is called Goursat's lemma.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

Bézout's identity ( also called Bezout's lemma ) is a theorem in the elementary theory of numbers: let a and b be integers, not both zero, and let d be their greatest common divisor.

Bézout's lemma is true in any principal ideal domain, but there are integral domains in which it is not true.

Bézout's lemma is a consequence of the Euclidean division defining property, namely that the division by a nonzero integer b has a remainder strictly less than | b |.

The Yoneda lemma is one of the most famous basic results of category theory ; it describes representable functors in functor categories.

One of these, Itō's lemma, expresses the composite of an Itō process ( or more generally a semimartingale ) dX < sub > t </ sub > with a twice-differentiable function f. In Itō's lemma, the derivative of the composite function depends not only on dX < sub > t </ sub > and the derivative of f but also on the second derivative of f. The dependence on the second derivative is a consequence of the non-zero quadratic variation of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

Their intersection is, which can be shown to be non-context-free by the pumping lemma for context-free languages.

* To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages or a number of other methods, such as Ogden's lemma, Parikh's theorem, or using closure properties.

In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring ; every division ring arises in this fashion from some simple module.

One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i. e. given any a and b, with a > b, there exist c and d, all positive and rational, such that

Euclid's classical lemma can be rephrased as " in the ring of integers every irreducible is prime ".

Weak König's lemma is provable in ZF, the system of Zermelo – Fraenkel set theory without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF.

However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.

All the different forms of the same verb constitute a lexeme and the form of the verb that is conventionally used to represent the canonical form of the verb ( one as seen in dictionary entries ) is a lemma.

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 ( Moore 1982: 168 ).

According to his lemma, a group of four manuscripts including Codex Monacensis 1086 are copies directly from the original.

* Lagrange's theorem ( group theory ) or Lagrange's lemma is an important result in Group theory

( which concerns the question of bounding the size of a group if there are fixed bounds both on the order of all of its elements and the number of elements needed to generate it ) and for Burnside's lemma ( a formula relating the number of orbits of a permutation group acting on a set with the number of fixed points of each of its elements ) though the latter had been discovered earlier and independently by Frobenius and Cauchy.

Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects.

One of Gelfand's original applications ( and one which historically motivated much of the study of Banach algebras ) was to give a much shorter and more conceptual proof of a celebrated lemma of Norbert Wiener ( see the citation below ), characterizing the elements of the group algebras L < sup > 1 </ sup >( R ) and whose translates span dense subspaces in the respective algebras.

It follows from Schur's lemma that the irreducible complex representations of an abelian group are all 1-dimensional.

is the direct limit of the GL ( n ), which embeds in GL ( n + 1 ) as the upper left block matrix, and the commutator subgroup agrees with the group generated by elementary matrices E ( A )= GL ( A ), by Whitehead's lemma.

In grasses, each floret ( flower ) is enclosed in a pair of papery bracts, called the lemma ( lower bract ) and palea ( upper bract ), while each spikelet ( group of florets ) has a further pair of bracts at its base called glumes.

In 1953, he founded a pioneering algebraic-computational linguistic group, and in 1961 he contributed to the proof of the pumping lemma for context-free languages ( sometimes called the Bar-Hillel lemma ).

In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Julius Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice.

In mathematics, Schreier's lemma is a theorem in group theory used in the Schreier – Sims algorithm and also for finding a presentation of a subgroup.

* The Ping-pong lemma, a useful way to exhibit a group as a free product

** Burnside's lemma, a theorem of group theory

Equivalently, a group is quasisimple if it is isomorphic to its commutator subgroup and its inner automorphism group Inn ( G ) ( its quotient by its center ) is simple ; due to Grün's lemma, Inn ( G ) must be non-abelian.

Some lexemes have several stems but one lemma.

Under this correspondence, Dickson's lemma may be seen as a special case of Hilbert's basis theorem stating that every polynomial ideal has a finite basis, for the ideals generated by monomials.

If is a unique factorization domain with field of fractions, then by Gauss's lemma is irreducible in, whether or not it is primitive ( since constant factors are invertible in ); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of.

In mathematics, more specifically modern algebra and commutative algebra, Nakayama's lemma also known as the Krull – Azumaya theorem governs the interaction between the Jacobson radical of a ring ( typically a commutative ring ) and its finitely generated modules.

Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field.

Nakayama's lemma makes precise one sense in which finitely generated modules over a commutative ring are like vector spaces over a field.

: Nakayama's lemma: Let U be a finitely generated right module over a ring R. If U is a non-zero module, then U · J ( R ) is a proper submodule of U.

In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it.

The significance of this lemma was recognized by Émile Borel ( 1895 ), and it was generalized to arbitrary collections of intervals by Pierre Cousin ( 1895 ) and Henri Lebesgue ( 1904 ).

The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ: in the case of an infinite dimensional space V, it depends on Zorn's lemma, one formulation of the axiom of choice.

Software limitations may result in its display either in full-sized capitals ( RUN ) or in full-sized capitals of a smaller font ; either is anyway regarded as an acceptable substitute for genuine small caps .</ ref > A related concept is the lemma ( or citation form ), which is a particular form of a lexeme that is chosen by convention to represent a canonical form of a lexeme.

In mathematics, a lemma ( plural lemmata or lemmas ) from the Greek λῆμμα ( lemma, “ anything which is received, such as a gift, profit, or a bribe ”) is a proven proposition which is used as a stepping stone to a larger result rather than as a statement of interest by itself.

Zorn's lemma was first discovered by Kazimierz Kuratowski in 1922, and then independently by Zorn in 1935.

Proof: If there is no possible move, then the lemma is vacuously true ( and the first player loses the normal play game by definition ).

But p is coprime to q and therefore to q < sup > n </ sup >, so by ( the generalized form of ) Euclid's lemma it must divide the remaining factor a < sub > 0 </ sub > of the product.

Should there be a nontrivial factor dividing all the coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense of Gauss's lemma ; this does not alter the set of rational roots and only strengthens the divisibility conditions.

On the other hand, by the Riemann – Lebesgue lemma, the weak limit exists and is zero.

The proof of Yoneda's lemma is indicated by the following commutative diagram:

In the case of distinguishing between two models, each of which has no unknown parameters, use of the likelihood ratio test can be justified by the Neyman – Pearson lemma, which demonstrates that such a test has the highest power among all competitors.

The origin of her name is believed by some like Robert S. P. Beekes to be Pre-Greek and related to pēnelops ( πηνέλοψ ) or * pēnelōps (* πηνέλωψ ), glossed by Hesychius as " some kind of bird " ( today arbitrarily identified with the Eurasian Wigeon, to which Linnaeus gave the binomial Anas penelope ), where-elōps (- έλωψ ) is a common pre-Greek suffix for predatory animals ;< ref > Zeno. org lemma relating πηνέλωψ ( gen. πηνέλοπος ) and < χην ( ά ) λοπες >· ὄρνεα ( predators ) ποιά.

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.