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Frobenius's theorem characterizes bipartite graphs that have perfect matching. Hall's theorem contains a characterization of bipartite graphs that have a matching from A to B. Koenig's theorem gives a formula for the matching number in a bipartite graph.
Frobenius's theorem establishes a connection between invaluability and integrability of a system of linearly independent vectors.
Frobenius's theorem has been completely proven.
Frobenius's theorem and adjustment The main field / C plays the role of unity in this case, since A K - A for any algebra A. Finally, Theorem 3.1 shows that the inverse algebra A is, indeed, up to matrices, the inverse of the algebra A in the sense of this operation All this allows us to define the structure of the group on the set of isomorphism classes of central bodies as follows.
Frobenius' Theorem 1.43 originally appeared as a theorem about the nature of solutions of certain systems of homogeneous linear equations with first order partial derivatives; see Frobenius and the discussion of invariants in §2.1. Its development into a theorem from differential geometry first occurred in Chevalley's important book on Lie groups. This book was collected together for the first time most of modern definitions and theorems on this subject. Subsequently it was further generalized - see Sussmann - but there is still a lot of work left, in particular on elucidating the structure of singular sets. In these and other works, the terms distribution or differential system apply to what we simply call a system of vector fields.
The Frobenius and Schur theorems have complex combinatorial proofs.
From the Frobenius theorem it follows that Frobenius groups are splittable. If H is an additional factor of the Frobenyus group, then the normalizer of any subgroup Yx of H is contained in the latter. Since the same is true for any subgroup conjugate to H, the invariant factor of the Frobenius group is strongly isolated. Consequently, any nonidentity element not contained in an invariant factor induces a regular automorphism in it.
According to the Frobenius-Perron theorem, any positive matrix (or non-negative, but indecomposable) has a positive real eigenvalue A mas, to which there corresponds a unique (up to a factor) eigenvector with positive components. Thus, the existence of a vector of priorities (weights of elements) is ensured in all cases when the matrix of judgments contains only positive elements.
According to the Frobenius theorem, all numbers (129) are different from zero and one sign.
By the Frobenius theorem [1, § 10, 9J, the seemingly more general case dwj i /, Λ Wk is reduced to that just considered using suitable linear combinations, and these conditions are necessary and sufficient for local integrability. They guarantee that a surface element can be extended from the infinitesimal to the local level; the question is about the possibility of continuing global level remains open. In this case N is characterized vector field X T 1, and, as shown in Section 2.3, integral curves always exist locally in X. IN general case n-dimensional submanifolds are invariant under local flows Фх generated by a vector field X satisfying the condition (wj Х) 0, and even locally generated if Фх can act on a point.
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Let be a body containing a body as a subbody R (\displaystyle \mathbb (R) ) real numbers, and two conditions are met:
In other words, L (\displaystyle \mathbb (L) ) is a finite-dimensional division algebra over the field of real numbers.
The Frobenius theorem states that any such body L (\displaystyle \mathbb (L) ):
Note that Frobenius's theorem applies only to finite-dimensional extensions R (\displaystyle \mathbb (R) ). For example, it does not cover the field of hyperreal numbers of non-standard analysis, which is also an extension R (\displaystyle \mathbb (R) ), but not finite-dimensional. Another example is the algebra of rational functions.
Consequences and remarks
The last three statements form the so-called generalized theorem Frobenius.
Division algebras over the field of complex numbers
Algebra of dimension n over the field complex numbers is an algebra of dimension 2n above R (\displaystyle \mathbb (R) ). The skew field of quaternions is not an algebra over a field C (\displaystyle \mathbb (C) ), since the center H (\displaystyle \mathbb (H) ) is a one-dimensional real space. Therefore, the only finite-dimensional division algebra over C (\displaystyle \mathbb (C) ) is algebra C (\displaystyle \mathbb (C) ).
Frobenius hypothesis
The theorem contains an associativity condition. What happens if you refuse this condition? The Frobenius conjecture states that even without the associativity condition for n other than 1, 2, 4, 8, in a real linear space Rn it is impossible to determine the structure of division algebra. The Frobenius hypothesis was proven in the 60s. XX century.
If at n>1 in space Rn bilinear multiplication without zero divisors is defined, then on the sphere S n-1 exists n-1 linearly independent vector fields. From the results obtained by Adams about the quantity vector fields on the sphere, it follows that this is only possible for spheres S 1 , S 3 , S 7. This proves the Frobenius conjecture.
see also
Literature
- Bakhturin Yu. A. Basic structures of modern algebra. - M.: Nauka, 1990. - 320 p.
- Kurosh A. G. Lectures on general algebra. 2nd ed. - M.: Nauka, 1973. - 400 p.
- Pontryagin L. S. Generalizations of numbers. - M.: Nauka, 1986. - 120 p. - (Library “Quantum”, issue 54).
Counting
setsReal numbers
and their extensionsExtension Tools
number systemsHierarchy of numbers − 1 , 0 , 1 , … (\displaystyle -1,\;0,\;1,\;\ldots ) If I = f0g, then F = R.
VII.6. Proof Frobenius' theorem
If I = f0g, then F = R.
If the dimension subspaces I is equal to 1, then F = C.
VII.6. Proof Frobenius' theorem
If I = f0g, then F = R.
If the dimension subspaces I equals 1, then F = C. Let the dimension subspaces I more than 1.
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
space I. Let i = p1 u. Then
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
i2 =
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u 2 (u2 ) =
i2 = p1 u 2 u
p 1 u 2 u =
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u 2 (u2 ) = 1:
i2 = p1 u 2 u
p 1 u 2 u =
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i = p1 u. Then i2 = 1:
By the sum i v = + x, where 2 R, x 2 I.
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
Lemma on the decomposition of elements from F
i v = + x, where
2 R, x 2 I. According to
(i + v) 2 I, in
in particular, (i + v)2< 0.
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
Lemma on the decomposition of elements from F
i v = + x, where
2 R, x 2 I. According to
(i + v) 2 I, in
in particular, (i + v)2< 0.
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
Lemma on the decomposition of elements from F
i v = + x, where
2 R, x 2 I.
According to
(i + v) 2 I,
in particular, (i + v)2< 0.
(i + v)2
(i + v)!
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
Lemma on the decomposition of elements from F
i v = + x, where
2 R, x 2 I.
According to
(i + v) 2 I,
in particular, (i + v)2< 0.
(i + v)2
(i + v)!
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
about decomposition
elements from
i v = + x, where
2 R, x 2 I.
(i + v). We have j2 = 1,
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
about decomposition
elements from
i v = + x, where
2 R, x 2 I.
(i1 + v). We have j2 = 1,
(i + v)2
i j = i
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
on the decomposition of elements
i v = + x, where
x 2 I .
(i1 + v). We have j2 = 1,
(i + v)2
i j = i
(i + v)2
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
about decomposition
elements
i v = + x, where
x 2 I .
(i1 + v). We have j2 = 1,
(i + v)2
i j = i
(i + v)2
(i + v)2
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
about decomposition
elements
i v = + x, where
x 2 I .
(i1 + v). We have j2 = 1,
(i + v)2
i j = i
(i + v)2
(i + v)2
x 2 I:
(i + v)2
(i + v)2
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
about decomposition
elements from
i v = + x, where
2 R, x 2 I.
(i + v)2
Means, ,
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
about decomposition
elements from
i v = + x, where
2 R, x 2 I.
(i + v). We have j2 = 1, i j 2I :
(i + v)2
I + j + i j ; ; ; 2R
body of quaternions.
VII.6. Proof Frobenius' theorem
Let the dimension subspaces I more than 1.
Let us take a linearly independent system of vectors fu; vg linear
space I. Let i =
u. Then i2 = 1:
about decomposition
elements from
i v = + x, where
2 R, x 2 I.
(i + v). We have j2 = 1, i j 2I :
(i + v)2
This means, by the lemma about embedding the body of quaternions in F,
I + j + i j ; ; ; 2R
body of quaternions.
Thus, if linear space I has dimension 3, then F is the body of quaternions.
VII.6. Proof Frobenius' theorem
subspaces I more than 3.
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I
Let's take linearly independent
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
x; y; z 2 I:
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
By virtue of the lemma on the decomposition of elements from F into the sum
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
By virtue of the lemma on the decomposition of elements from F into the sum
x; y; z 2 I:
By virtue of lemmas on subspace I t = m + i + j + k 2I . From linear independence fi vector systems; j; k; mg following
it blows that t 6= 0.
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
By virtue of the lemma on the decomposition of elements from F into the sum
x; y; z 2 I:
Subspace Lemma I
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
By virtue of the lemma on the decomposition of elements from F into the sum
x; y; z 2 I:
It has been proven that 0 6= t = m + i + j + k 2 I . By Subspace Lemma I
i t = i m + k j =
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
By virtue of the lemma on the decomposition of elements from F into the sum
x; y; z 2 I:
It has been proven that 0 6= t = m + i + j + k 2 I . By Subspace Lemma I
i t = i m + k j = x + k j
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
By virtue of the lemma on the decomposition of elements from F into the sum
x; y; z 2 I:
It has been proven that 0 6= t = m + i + j + k 2 I . By Subspace Lemma I
i t = i m + k j = x + k j 2 I:
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
By virtue of the lemma on the decomposition of elements from F into the sum
Similarly, we can prove that j t 2 I, k t 2 I.
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
By virtue of the lemma on the decomposition of elements from F into the sum
x; y; z 2 I:
It has been proven that
0 6= t = m + i + j + k 2 I . Polemma about subpro-
travel I
i t 2 I, j t 2 I,
Let's put n =
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
We found n 2 I such that n2 = 1, 0 6= i n 2 I, 0 6= j n 2 I,
By the lemma on the embedding of the body of quaternions in F
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
We found n 2 I such that n2 = 1, 0 6= i n 2 I, 0 6= j n 2 I,
By the lemma on the embedding of the body of quaternions in F
i n = n i; j n = n j; k n = n k:
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
We found n 2 I such that n2 = 1, 0 6= i n 2 I, 0 6= j n 2 I,
By the lemma on the embedding of the body of quaternions in F
i n = n i; j n = n j; k n = n k:
N i j = i n j =
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
We found n 2 I such that n2 = 1, 0 6= i n 2 I, 0 6= j n 2 I,
By the lemma on the embedding of the body of quaternions in F
i n = n i; j n = n j; k n = n k:
N k = n i j = i n j =
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
We found n 2 I such that n2 = 1, 0 6= i n 2 I, 0 6= j n 2 I,
By the lemma on the embedding of the body of quaternions in F
i n = n i; j n = n j; k n = n k: k n = n k = n i j = i n j =
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
We found n 2 I such that n2 = 1, 0 6= i n 2 I, 0 6= j n 2 I,
By the lemma on the embedding of the body of quaternions in F
i n = n i; j n = n j; k n = n k:
k n = n k = n i j = i n j = i (j n) =
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
We found n 2 I such that n2 = 1, 0 6= i n 2 I, 0 6= j n 2 I,
By the lemma on the embedding of the body of quaternions in F
i n = n i; j n = n j; k n = n k:
VII.6. Proof Frobenius' theorem
It remains to consider the case when the dimension subspaces I greater than 3. We have proven that then F includes the field of quaternions.
Let's take linearly independent system of vectors fi; j; k; mg, where i2 = j2 = k2 = 1, i j = j i = k, j k = k j = i, k i = i k = j.
We found n 2 I such that n2 = 1, 0 6= i n 2 I, 0 6= j n 2 I,
By the lemma on the embedding of the body of quaternions in F
i n = n i; j n = n j; k n = n k:
k n = n k = n i j = i n j = i (j n) = k n:
Therefore, 2k n = 0, a contradiction.
VII. Frobenius's theorem
Theorem 2. Let F be a body, and R F ,
9i1; i2 ; : : : ; in
9 0 ;1 ;2 ; : : : ;n 2 R
z = 0 +1 i1 +2 i2 + : : : +n in :
Then F is either R, C, or a quaternion body.
The theorem has been proven.
attention!
e-mail: [email protected]; [email protected]
websites: http://melnikov.k66.ru; http://melnikov.web.ur.ru
FROBENIUS THEOREM
Describing all finite-dimensional associative real algebras without zero divisors, proven by G. Frobenius. F. T. states that:
1) Field real numbers and complex numbers are the only finite-dimensional real associative-commutative algebras without zero divisors.
2) The skew field of quaternions is the only finite-dimensional real associative but not commutative algebra without zero divisors.
There is also a description of alternative finite-dimensional algebras without zero divisors:
3) Cayley algebra is the only finite-dimensional real alternative, but not associative algebra without zero divisors.
The combination of these three statements is cash. generalized Frobenius theorem. All algebras involved in the formulation of the theorem turn out to be algebras with by unambiguous division and with one. F. t. cannot be generalized to the cases of non-alternative algebras. It has been proven, however, that any finite-dimensional real algebra without zero divisors can only take values equal to 1, 2, 4 or 8.Lit.: Frobenius F., "J. reine und angew. Math.", 1877, Bd 82, S. 230-315; Kurosh A.G., Lectures on general algebra, 2nd ed., M., 1973.
O. A. Ivanova.Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.
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Books
- , Zorich V.. This book is a record of an annual experimental special course in natural science content for mathematicians, as well as students and specialists in other specialties. It presents three topics: -…
- Mathematical analysis of problems in natural science, V. A. Zorich. This book is a record of a one-year experimental special course in natural science content for mathematicians, as well as students and specialists in other specialties. It presents three topics: -…