A note on a theorem of Armand Borel
... Proof of Theorem 2. We recall that F = A(y1, ...,ym) means that the monomials ViiVu-'-yik' h < *2 < ••• < *fc(& = l>2,...,m), together with the unit element form an additive base for the vector space F over K. This is more general than an exterior algebra, since it may happen that y\ 4= 0, as, for e ...
... Proof of Theorem 2. We recall that F = A(y1, ...,ym) means that the monomials ViiVu-'-yik' h < *2 < ••• < *fc(& = l>2,...,m), together with the unit element form an additive base for the vector space F over K. This is more general than an exterior algebra, since it may happen that y\ 4= 0, as, for e ...
08. Zermelo-Fraenkel (ZF) Formal Set Theory
... Comment: This axiom says that for any set x other than the empty set, there is a “minimal” member y of x that has no members in common with members of x. This rules out circular chains of sets (e.g., x ∈ y and y ∈ z and z ∈ x) and infinitely descending chains of sets. In particular, it rules out the ...
... Comment: This axiom says that for any set x other than the empty set, there is a “minimal” member y of x that has no members in common with members of x. This rules out circular chains of sets (e.g., x ∈ y and y ∈ z and z ∈ x) and infinitely descending chains of sets. In particular, it rules out the ...
Math 236H Final exam
... modulo p. The only number that fits this description is 1. Hence the p-Sylow subgroup is normal, so there is a homomorphism φ1 from G onto the quotient group, which must be Cq . Similarly, there is only one q-Sylow subgroups and hence a homomorphism φ2 to Cp . Using these we get a homomorphism to Cp ...
... modulo p. The only number that fits this description is 1. Hence the p-Sylow subgroup is normal, so there is a homomorphism φ1 from G onto the quotient group, which must be Cq . Similarly, there is only one q-Sylow subgroups and hence a homomorphism φ2 to Cp . Using these we get a homomorphism to Cp ...
Document
... Problem 2. Mark each of the following true or false (no explanation is required) 1. Every group is isomorphic to some group of permutations. (yes, by Cayley’s Theorem) 2. Every permutation is a one-to-one function. (yes, by definition of permutation) 3. Every group G is isomorphic to a subgroup of S ...
... Problem 2. Mark each of the following true or false (no explanation is required) 1. Every group is isomorphic to some group of permutations. (yes, by Cayley’s Theorem) 2. Every permutation is a one-to-one function. (yes, by definition of permutation) 3. Every group G is isomorphic to a subgroup of S ...
Trivial remarks about tori.
... torus over C and L = X ∗ (T ) then for any abelian topological group W (for example, C× , or R ) there’s a canonical bijection between Π := Hom(Hom(L, W ), C× ) and R := Hom(W, Hom(L̂, C× )) (all homs are continuous group homs). So if W = k × for k a topological field, one sees that Hom(T (k), C× ) ...
... torus over C and L = X ∗ (T ) then for any abelian topological group W (for example, C× , or R ) there’s a canonical bijection between Π := Hom(Hom(L, W ), C× ) and R := Hom(W, Hom(L̂, C× )) (all homs are continuous group homs). So if W = k × for k a topological field, one sees that Hom(T (k), C× ) ...
Rings of Fractions
... Theorem 49. Let R be a commutative ring. Let D be any nonempty subset of R that does not contain 0, does not contain any zero divisors, and is closed under multiplication. Then there exists a commutative ring Q with 1 such that Q contains R as a subring and every element of D is a unit in Q. Theorem ...
... Theorem 49. Let R be a commutative ring. Let D be any nonempty subset of R that does not contain 0, does not contain any zero divisors, and is closed under multiplication. Then there exists a commutative ring Q with 1 such that Q contains R as a subring and every element of D is a unit in Q. Theorem ...
![FINITE POWER-ASSOCIATIVE DIVISION RINGS [3, p. 560]](http://s1.studyres.com/store/data/013411645_1-e3a70dd07d74f412121f6eca2ee1a3db-300x300.png)
FINITE POWER-ASSOCIATIVE DIVISION RINGS [3, p. 560]
... in generality in restricting ourselves to algebras. An algebra is a division algebra if left and right multiplications by a nonzero element are bijections; for finite-dimensional algebras this is equivalent to the nonexistence of proper zero divisors. Following N. Jacobson, we define a Jordan divisi ...
... in generality in restricting ourselves to algebras. An algebra is a division algebra if left and right multiplications by a nonzero element are bijections; for finite-dimensional algebras this is equivalent to the nonexistence of proper zero divisors. Following N. Jacobson, we define a Jordan divisi ...
Summary of Chapter 15, Quotient Groups
... Theorem 2: Let H be a normal subgroup of group G. If Ha = Hc and Hb = Hd , then H (ab) = H (cd ) . (Thus, coset multiplication is uniquely defined for normal subgroups.) ...
... Theorem 2: Let H be a normal subgroup of group G. If Ha = Hc and Hb = Hd , then H (ab) = H (cd ) . (Thus, coset multiplication is uniquely defined for normal subgroups.) ...
Chapter 3 – Group Theory – p. 1
... • Without proof (see e.g. F. A. Cotton): ( ABC )−1 = A−1 B −1C −1 . (3.1: Example for a group) ...
... • Without proof (see e.g. F. A. Cotton): ( ABC )−1 = A−1 B −1C −1 . (3.1: Example for a group) ...
p. 1 Math 490 Notes 4 We continue our examination of well
... empty set φ is a well-ordered set (vacuously), and the ordinal containing φ is naturally denoted 0 (zero). Now consider all well-ordered sets with exactly n elements for some n ∈ N. It should be easy to see that all such well-ordered sets are similar to each other, and thus they all belong to the sa ...
... empty set φ is a well-ordered set (vacuously), and the ordinal containing φ is naturally denoted 0 (zero). Now consider all well-ordered sets with exactly n elements for some n ∈ N. It should be easy to see that all such well-ordered sets are similar to each other, and thus they all belong to the sa ...
INTRODUCTORY GROUP THEORY AND FERMAT`S LITTLE
... check to see if G is closed by verifying that the product of any elements in G are not equivalent to 0 mod p (meaning that no divisors of p are in G). We know that p is prime, hence its only divisors are 1 and p itself. Therefore, there is no way that the product of any two elements to be equal to 0 ...
... check to see if G is closed by verifying that the product of any elements in G are not equivalent to 0 mod p (meaning that no divisors of p are in G). We know that p is prime, hence its only divisors are 1 and p itself. Therefore, there is no way that the product of any two elements to be equal to 0 ...
Another type of data structure: a list. List elements may be of different
... Load the data from a file exercices6.Rdata Check, how many genes have genes1 and genes2 in common. How many genes do they contain in total? How many genes are present on genes1 and not on genes2 and vice ...
... Load the data from a file exercices6.Rdata Check, how many genes have genes1 and genes2 in common. How many genes do they contain in total? How many genes are present on genes1 and not on genes2 and vice ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.