The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by F. G. Frobenius and Issai Schur.

* Frobenius also has proved the following fundamental theorem: If a positive integer n divides the order | G | of a finite group G, then the number of solutions of the equation x < sup > n </ sup > = 1 in G is equal to kn for some positive integer k. He also posed the following problem: If, in the above theorem, k = 1, then the solutions of the equation x < sup > n </ sup > = 1 in G form a subgroup.

A group G is said to be a Frobenius group if there is a subgroup H < G such that

The theorem of Frobenius states that for any given choice of Π the primes p for which the splitting type of P mod p is Π has a natural density δ, with δ equal to the proportion of g in G that have cycle type Π.

The statement of the more general Chebotarev theorem is in terms of the Frobenius element of a prime ( ideal ), which is in fact an associated conjugacy class C of elements of the Galois group G. If we fix C then the theorem says that asymptotically a proportion | C |/| G | of primes have associated Frobenius element as C. When G is abelian the classes of course each have size 1.

Since G is compact in this topology, there is a unique Haar measure μ on G. For every prime v of K not in S there is an associated Frobenius conjugacy class F < sub > v </ sub >.

A consequence of this version of the theorem is that the Frobenius elements of the unramified primes of L are dense in G.

The general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900.

The normalizers of these maximal abelian subgroups turn out to be exactly the maximal proper subgroups of G. These normalizers are Frobenius groups whose character theory is reasonably transparent, and well-suited to manipulations involving character induction.

They are named after F. G. Frobenius.

The subgroup H of a Frobenius group G fixing a point of the set X is called the Frobenius complement.

) The Frobenius group G is the semidirect product of K and H:

The Frobenius kernel K is uniquely determined by G as it is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus theorem.

* 1917 – Ferdinand Georg Frobenius, German mathematician ( b. 1849 )

* In the theory of numerical semigroups, the conductor of a numerical semigroup ( also called the Frobenius number of the semigroup ) is the greatest integer not in the semigroup.

Frobenius theorem: The only finite-dimensional division algebras over the reals are the reals themselves, the complex numbers, and the quaternions.

The finite fields are classified by size ; there is exactly one finite field up to isomorphism of size p < sup > k </ sup > for each prime p and positive integer k. Each finite field of size q is the splitting field of the polynomial x < sup > q </ sup > − x, and thus the fixed field of the Frobenius endomorphism which takes x to x < sup > q </ sup >.

* 1849 – Ferdinand Georg Frobenius, German mathematician ( d. 1917 )

He continued with his study in Berlin ( 1899 / 1900 ) under the German mathematicians Frobenius and Fuchs and in Göttingen ( 1900 / 1901 ) under Klein and Hilbert.

In 1912 he started studies in mathematical physics at the University of Budapest, with summer visits to the University of Berlin and the University of Göttingen, where he attended lectures by Frobenius and Hilbert, amongst others.

In 1730, German chemist Frobenius gave this liquid its present name, ether, which is Greek for “ heavenly .” But 112 more years would pass before ether ’ s anesthetic powers were fully appreciated.

( Closeness can be measured by any matrix norm invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.

Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8. 28659 instead of the minimum 8. 12404.

The Frobenius inner product, sometimes denoted A: B is the component-wise inner product of two matrices as though they are vectors.

This inner product induces the Frobenius norm.

The chapel is noted for its Frobenius organ in the west gallery.

The closeness of fit is measured by the Frobenius norm of.

where denotes the Frobenius norm.

The church houses two organs: the oldest in the west pulpit was built by Demant & Son in 1861 with a baroque facade from 1633 ; the other from 1974 is by Frobenius & Sons.

# ( That is, the Frobenius norm of A can be computed by the eigenvalues of A.

One method of improving efficiency further in some cases is the Frobenius pseudoprimality test ; a round of this test takes about three times as long as a round of Miller – Rabin, but achieves a probability bound comparable to seven rounds of Miller – Rabin.

The Perron – Frobenius theorem ensures that every stochastic matrix has such a vector, and that the largest absolute value of an eigenvalue is always 1.

The description is in terms of Frobenius elements, and generalises in a far-reaching way the quadratic reciprocity law that gives full information on the decomposition of prime numbers in quadratic fields.

Grothendieck proved an analogue of the Lefschetz fixed point formula for l-adic cohomology theory, and by applying it to the Frobenius automorphism F was able to prove the following formula for the zeta function.

The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks.

* This estimate can be used to find the absolute value of any eigenvalue α of Frobenius on a fiber of E as follows.

" Aimé Césaire also quoted Frobenius as praising African people as being " civilized to the marrow of their bones ", as opposed to the degrading vision encouraged by colonial propaganda.

Another theme sees Ecbatana, the seven-walled " city of Dioce ", blend with the city of Wagadu, from the African tale of Gassire's Lute that Pound derived from Frobenius.

Whereas in the CN-case, the resulting maximal subgroups M are still Frobenius groups, the maximal subgroups that occur in the proof of the odd-order theorem need no longer have this structure, and the analysis of their structure and interplay produces 5 possible types of maximal subgroups, called types I, II, III, IV, V. Type I subgroups are of " Frobenius type ", a slight generalization of Frobenius group, and in fact later on in the proof are shown to be Frobenius groups.

For real-analytic J, the Newlander – Nirenberg theorem follows from the Frobenius theorem ; for ( and less smooth ) J, analysis is required ( with more difficult techniques as the regularity hypothesis weakens ).

For loan to the Government of India under subsection ( G ) of Section 104 of the Act, the rupee equivalent of not more than $538 million for financing such projects to promote balanced economic development as may be mutually agreed.

With respect to Article 2,, paragraphs 1 ( B ) and 1 ( C ): Uses of Section 104 ( E ) and Section 104 ( G ) rupees: The Government of India will use the amount of rupees granted or loaned to it by the United States pursuant to paragraphs 1 ( B ) and 1 ( C ) for projects to promote economic development with emphasis upon the agricultural sector including food reserve storage structures and facilities as may from time to time be agreed upon by the authorized representatives of the United States and the authorized representatives of the Government of India, in the following sectors: A.

Other economic development projects consistent with the purposes of Sections 104 ( E ) and 104 ( G ) of the Act.

Conversely, let G be any polynomial such that Af.

It is easily seen that Af divides G.

A function G such that Af, i.e., Af, must be a polynomial function of degree Af or less: Af.

G.

The anode potential codetermines the gain, G, and magnification, M, of the stage.

The angle generated by the platform servo **yf multiplied by G is the effective acceleration acting on the accelerometer.

They found it at the junction between Troops H and G, and prepared to counterattack.

* Lloyd, G. E. R. ( 1968 ).

Although there are seven other types of annual awards presented by the Academy ( the Irving G. Thalberg Memorial Award, the Jean Hersholt Humanitarian Award, the Gordon E. Sawyer Award, the Scientific and Engineering Award, the Technical Achievement Award, the John A. Bonner Medal of Commendation, and the Student Academy Award ) plus two awards that are not presented annually ( the Special Achievement Award in the form of an Oscar statuette and the Honorary Award that may or may not be in the form of an Oscar statuette ), the best known one is the Academy Award of Merit more popularly known as the Oscar statuette.

* Wickens, G. M.

After this Confederate defeat at the Battle of Mill Springs, Davis sent Johnston a brigade and a few other scattered reinforcements, and he sent Gen. P. G. T.

Among his staff was Isham G. Harris, the Governor of Tennessee, who had ceased to make any real effort to function as governor after learning that Abraham Lincoln had appointed Andrew Johnson as military governor of Tennessee.