Theorem 1. 4: The identity element of a group is unique.

Theorem 1. 6: For all elements in a group.

Theorem 1. 7: For all elements and in group,.

Theorem 1. 8: For all elements in a group, then.

Theorem 1. 3: For all elements in a group, there exists a unique such that, namely.

:: Theorem: A finitely presented residually finite group has solvable word problem.

If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal.

( There is a fundamental theorem holding in every finite group, usually called Fermat's little Theorem because Fermat was the first to have proved a very special part of it.

* Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group ( Mordell's Theorem, later generalized to the Mordell – Weil theorem ).

Then Hahn's Embedding Theorem reduces to Hölder's theorem ( which states that a linearly ordered abelian group is Archimedean if and only if it is a subgroup of the ordered additive group of the real numbers ).

The corresponding result for the multiplicative group is known as Hilbert's Theorem 90, and was known before 1900.

: Theorem: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G / N is simple.

For a proof of the first part of the Latin square property, see the Wikipedia page on elementary group theory ( Theorem 1. 3 ).

He was appointed a lecturer in mathematics at Cambridge in 1927, where his 1935 lectures on the Foundations of Mathematics and Gödel's Theorem inspired Alan Turing to embark on his pioneering work on the Entscheidungsproblem ( decision problem ) using a hypothetical computing machine.

" After completing his dissertation ( 1907-see Van Dalen ), Brouwer made a conscious decision to temporarily keep his contentious ideas under wraps and to concentrate on demonstrating his mathematical prowess " ( Davis ( 2000 ), p. 95 ); by 1910 he had published a number of important papers, in particular the Fixed Point Theorem.

Theorem: K is not a computable function.

: Turing's thesis: " Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i. e. by one of his machines, is equivalent to Church's thesis by Theorem XXX.

: Theorem on projections: Let the function f: X → B be such that a ~ b → f ( a )

The Second Main Theorem, more difficult than the first one tells that there are relatively few values which the function assumes less often than average.

" The Implicit Function Theorem states that if is defined on an open disk containing, where,, and and are continuous on the disk, then the equation defines as a function of near the point and the derivative of this function is given by ..."

Tautology -- Temporal logic -- Term -- Term logic -- Ternary logic -- Theorem -- Tolerance -- Trilemma -- Truth -- Truth condition -- Truth function -- Truth value -- Type theory

This theorem is called the Second Fundamental Theorem of Nevanlinna Theory, and it allows to give an upper bound for the characteristic function in terms of N ( r, a ).

The Second Fundamental Theorem implies that the set of deficient values of a function meromorphic in the plane is at most countable and the following relation holds:

Theorem For ƒ in L < sup > 1 </ sup >( R < sup > d </ sup >), the above series converges pointwise almost everywhere, and thus defines a periodic function Pƒ on Λ. Pƒ lies in L < sup > 1 </ sup >( Λ ) with || Pƒ ||< sub > 1 </ sub > ≤ || ƒ ||< sub > 1 </ sub >.

* Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud

* Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud

Theorem ( Dini's test ): Assume a function f satisfies at a point t that

However, unlike the value of most other assets, the value of land is largely a function of government spending on services and infrastructure ( a relationship demonstrated by economists in the Henry George Theorem ).

Schwarz's Theorem then provides us with a necessary criterion for the existence of a potential function.

The universality of the Riemann zeta function was first proven by Sergei Mikhailovitch Voronin in 1975 and is sometimes known as Voronin's Universality Theorem.

Theorem For any normalized continuous positive definite function f on G ( normalization here means f is 1 at the unit of G ), there exists a unique probability measure on such that

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

Problem II. 8 in the Arithmetica ( edition of 1670 ), annotated with Fermat's comment which became Fermat's Last Theorem.

This result is now known as Church's Theorem or the Church – Turing Theorem ( not to be confused with the Church – Turing thesis ).

For with coprime and, one can use the Prime-Factor ( Good-Thomas ) algorithm ( PFA ), based on the Chinese Remainder Theorem, to factorize the DFT similarly to Cooley – Tukey but without the twiddle factors.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

Using Rogers ' characterization of acceptable programming systems, Rice's Theorem may essentially be generalized from Turing machines to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the black-box behavior of computer programs.

Germain's best work was in number theory, and her most significant contribution to number theory dealt with Fermat's Last Theorem.

Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 ( see Correspondence with Legendre ).

* A space elevator is also constructed in the course of Clarke's final novel ( co-written with Frederik Pohl ), The Last Theorem.

:: Boone-Rogers Theorem: There is no uniform partial algorithm which solves the word problem in all finitely presented groups with solvable word problem.

Strangely, this approach is often used for cases where Theorem I applies, which creates problems with the basic model assumptions.

The play opens on 10 April 1809, in a garden front room of a country house in Derbyshire with tutor Septimus Hodge trying to distract his 13 year-old pupil Thomasina from her enquiries as to the meaning of a " carnal embrace " by challenging her to prove Fermat's Last Theorem so he can focus on reading the poem ' The Couch of Eros ', a piece written by another character, Mr. Ezra Chater.

Remembrance of the story of the bull's hide and the foundation of Carthage is preserved in mathematics in connection with the Isoperimetric problem which is sometimes called Dido's Problem ( and similarly the Isoperimetric theorem is sometimes called Dido's Theorem ).

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

, with some help from Richard Taylor, proved the Taniyama – Shimura – Weil conjecture for all semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

* Zenkov, DV, AM Bloch, and JE Marsden The Lyapunov-Malkin Theorem and Stabilization of the Unicycle with Rider.

Gödel's First Incompleteness Theorem then tells us that there are certain consistent bodies of propositions with no recursive axiomatization.

The main difficulty in verifying Perelman's proof of the Geometrization conjecture was a critical use of his Theorem 7. 4 in the preprint ' Ricci Flow with surgery on three-manifolds '.

Under Desargues ' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically.

Gentzen's so-called " Main Theorem " ( Hauptsatz ) about LK and LJ was the cut-elimination theorem, a result with far-reaching meta-theoretic consequences, including consistency.

Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble ( i. e., the total or convective time derivative is zero ).

Corollary ( Pointwise Ergodic Theorem ): In particular, if T is also ergodic, then is the trivial σ-algebra, and thus with probability 1:

Moreover, as long as the polynomial factors at each stage are relatively prime ( which for polynomials means that they have no common roots ), one can construct a dual algorithm by reversing the process with the Chinese Remainder Theorem.