ENTRY ABSTRACT ALGEBRA Authors: started Oliver Knill: September 2003 Literature: Lecture notes +------------------------------------------------------------ | additive +------------------------------------------------------------ A function f: G to H from a semigroup G to a semigroup H is additive if f(a+b) = f(a) + f(b). A group-valued function on sets is additive if f(Y cup Z) = f(Y) + f(Z) if Y and Z are disjoint. +------------------------------------------------------------ | algebra +------------------------------------------------------------ An algebra over a field K is a ring with 1 which is also a vector space over K and whose multiplication is bilinear with respect to K. Examples: the complex numbers C is an algebra over the field of real numbers K=R. The quaternion algebra H is an algebra over the field of complex numbers. The matrix algebra M(n,R) is an algebra over the field R. +------------------------------------------------------------ | An algebraic number field +------------------------------------------------------------ An algebraic number field is a subfield of the complex numbers that arises as a finite degree algebraic extension field over the field of rationals. +------------------------------------------------------------ | alternating group +------------------------------------------------------------ The alternating group G is the subgroup of the symmetric group of n objects given by the elements which can be written as a product of an even number of transpositions. +------------------------------------------------------------ | Artinian module +------------------------------------------------------------ An Artinian module is a module which satisfies the descending chain condition. Every Artinian module is a Noetherian module but the integers for example are a Noetherian module which is not an Artinian module. +------------------------------------------------------------ | Artinian ring +------------------------------------------------------------ An Artinian ring is a ring which when considered as a R-module is an Artinian module. +------------------------------------------------------------ | Artinian ring +------------------------------------------------------------ Two elements of an integral domain that are unit-multipliers of each other are called associate numbers. +------------------------------------------------------------ | Cayley's theorem +------------------------------------------------------------ Cayley's theorem assures that every finite group is isomorphic to a permutation group. +------------------------------------------------------------ | center +------------------------------------------------------------ The center of a group (G,*) is the set of all elements g which satisfy g h = h g for all h in G. The center is a subgroup of G. +------------------------------------------------------------ | commutator +------------------------------------------------------------ The commutator of two elements g,h in a group (G,*) is defined as g,h=g*h*g^-1*h^-1. +------------------------------------------------------------ | commutator subgroup +------------------------------------------------------------ The commutator subgroup of a group (G,*) is the set of all commutators g,h in G. It is a subgroup of G. +------------------------------------------------------------ | factor group +------------------------------------------------------------ A factor group G/N is defined when N is a normal subgroup of the group G. It is the group, where the elements are equivalent classes g N and operation (g N) (h N) = (g h) N which is defined because N was assumed to be normal. For example, if G is the group of additive integers and N=k N with an integer k, then G/N = Z_k is finite group of integers modulo k. +------------------------------------------------------------ | finite group +------------------------------------------------------------ A group is called a finite group if G is a set with finitely many elements. For example, the set of all permutations of a finite set form a finite group. The set of all operations on the Rubik cube form a finite group. +------------------------------------------------------------ | group +------------------------------------------------------------ A group (X,+,0) is a set X with a binary operation + and a zero element 0 (also called neutral element or identity) with the following properties (a+b)+c = a+(b+c) associativity a+0 = a zero element forall a exists b a+b=0 inverse Examples: the real numbers form a group under addition 5+2.34=7.34, 3-3=0. the set GL(n,R) of real matrices with nonzero determinant form a group under matrix multiplication the nonzero integers form a group under multiplication 4*7=28. all the invertible linear transformations of the plane plane form a group under composition. The "zero element" is the identity transformation T(x)=x. all the continuous functions on the unit interval form a group with addition (f+g)(x) = f(x)+g(x). all the permutations on a finite set form a group under composition. the set of subsets Y of a set X with the operation A Delta B = (A cup B) setminus (A cap B) form a group. The inverse of A is A itself because A Delta A = emptyset, the zero element is emptyset. +------------------------------------------------------------ | normal subgroup +------------------------------------------------------------ a normal subgroup of a group (G,*) is a subgroup (H,*) of (G,*) which has the property that for all g in H and all g in G one has g^-1 h g is in H. For an abelian group all subgroups are normal. The subgroup Sl(n,R) of Gl(n,R) is a normal subgroup. +------------------------------------------------------------ | ring +------------------------------------------------------------ A ring (X,+,*,0) is a set X with a binary operation + and a binary operation * such that (X,+,0) is a commutative group and (X,*) is a semigroup and such that the distributivity laws a*(b+c) = a*b + a*c, (a+b)*c - a*c+b*c hold. Examples: the integers Z form a ring with addition and multiplication the set of rational numbers Q, the set of real numbers R or the complext numbers C form a ring with addition and multiplication. the set of 3x3 matrices with real entries form a ring with addition and matrix multiplication. the set P of polynomials with real coefficients form a ring with addition and multiplication. the set of subsets Y of a set X with addition Delta and multiplication cap forms a ring. the set of continuous functions on an interval 0,1 with addition (f+g)(x) = f(x)+g(x) and multiplication f*g(x) = f(x) g(x). +------------------------------------------------------------ | commutative group +------------------------------------------------------------ A commutative group is a group (X,+,0) which is commutative: a+b=b+a. the set of real numbers R forms a commutative group under addition. the set of permutations S of a set X form a noncommutative group under composition. +------------------------------------------------------------ | commutative ring +------------------------------------------------------------ A commutative ring is a ring (X,+,*,0) for which the multiplicative semigroup (X,*) is commutative: a*b = b*a. Examples: the integers form a commutative ring. the set of 2 x 2 matrices form a noncommutative ring the set of polyomials with real coefficients (x^2+pi x+2) * (x+5x) = 6 x^3 + 6 pi x^2+12x. +------------------------------------------------------------ | function field +------------------------------------------------------------ A function field is a finite extension of the field C(z) of rational functions in the variable z. +------------------------------------------------------------ | homomorphism +------------------------------------------------------------ An homomorphism phi between two groups G,H is a map f: G to H which has the property phi(g*h) = phi(g) * phi(h) and phi(0)=0 for all elements g,h in G. Examples: if G is the multiplicative group (R^+,*) of positive real numbers and H is the additive group (R,+) of all positive real numbers then phi(x)=log(x) is a homomorphism: if G is the group of matrices with nonzero determinant and H is the group of nonzero real numbers and phi(A)= det(A), we have phi(x*y) = phi(x) phi(y). +------------------------------------------------------------ | isomorphism +------------------------------------------------------------ An isomorphism phi between two groups G,H is a homomorphism between groups which is also invertible. +------------------------------------------------------------ | number field +------------------------------------------------------------ A number field is a finite extension of Q, the field of rational numbers. It is a field extension of Q which is also a vector space of finite dimension over Q. Since the elements of a number field are algebraic numbers, roots of a fixed polyonomial a_0+a_1 z+... + z^n with integer coefficitients, one calls them also algebraic number fields. The study of algebraic number fields is part of algebraic number theory. Examples: quadratic fields: Q(sqrtd), where d is a rational number. It is in general a field extension of degree 2 over the field of rational number. cyclotimic fields: Q(xi), where xi is a n'th root of 1. It is a field extension of degree phi(n), where phi(n) is the Euler function. +------------------------------------------------------------ | octonions +------------------------------------------------------------ The octonions can be written as linear combinations of elements e_0,e_1,e_2,...,e_7. The multiplication is determined by the multiplication table * 1 e_1 e_2 e_3 e_4 e_5 e_6 e_7 1 1 e_1 e_2 e_3 e_4 e_5 e_6 e_7 e_1 e_1 -1 e_4 e_7 -e_2 e_6 -e_5 -e_3 e_2 e_2 -e_4 -1 e_5 e_1 -e_3 e_7 -e_6 e_3 e_3 -e_7 -e_5 -1 e_6 e_2 -e_4 e_1 e_4 e_4 e_2 -e_1 -e_6 -1 e_7 e_3 -e_5 e_5 e_5 -e_6 e_3 -e_2 -e_7 -1 e_1 e_4 e_6 e_6 e_5 -e_7 e_4 -e_3 -e_1 -1 e_2 e_7 e_7 e_3 e_6 -e_1 e_5 -e_4 -e_2 -1 Octonions are also called Cayley numbers. The multiplication of octonions is not associative. Octonions have been discovered by John T. Graves in 1843 and independently by Arthur Cayley. +------------------------------------------------------------ | order +------------------------------------------------------------ The order of a finite group is the set of elements in the group. +------------------------------------------------------------ | p-group +------------------------------------------------------------ A p-group is a finite group with order p^n, where p is a prime integer and n>0. +------------------------------------------------------------ | quaterions +------------------------------------------------------------ The quaterions can be written as linear combinations of elements 1,i,j,k. The multiplication is determined by the multiplication table * 1 i j k 1 1 i j k i i -1 k -j j j -k -1 i k k j -i -1 Quaternions are useful to compute rotations in three dimensions. +------------------------------------------------------------ | semigroup +------------------------------------------------------------ A semigroup (X,+) is a set X with a binary operation + which satisfies the associativity law (a+b)+c = a+(b+c). Examples: a group is a semigroup. the set of finite words in an alphabet with composition form a semigroup word1 + word2 = word1word2 the natural numbers form a semigroup under addition. +------------------------------------------------------------ | commutative semigroup +------------------------------------------------------------ A commutative semigroup is a semigroup (X,+) which is commutative. a+b=b+a. the natural numbers form a commutative semigroup under addition. composition of words over a finite alphabet form a noncommutative semigroup +------------------------------------------------------------ | kernel +------------------------------------------------------------ The kernel of a homomorphism between two groups G,H is the set of all elements in G which are maped to the zero element of H. For example, SL(n,R) is the kernel of the homomorphism from GL(n,R) to R setminus 0 defined by phi(A) = det(A). +------------------------------------------------------------ | subgroup +------------------------------------------------------------ A subgroup of a group G is a subset of G which is also a group. Examples: the set of n x n matrices with determinant 1 is a subgroup of the set of n x n matrices with nonzero determinant. the trivial subgroup 0 is always a subgroup of a group (G,*,0). +------------------------------------------------------------ | Theorem of Cauchy +------------------------------------------------------------ The Theorem of Cauchy in group theory states that every finite group whose order is divisible by a prime number p contains a subgroup of order p. +------------------------------------------------------------ | sedenions +------------------------------------------------------------ sedenions form a zero Divisor Algebra. By a theorem of Frobenius (1877), there are three and only three associative finite division algebras: the real numbers R, the complex numbers C and the quaternions Q. Similar algebras in higher dimensions have zero divisors: sedenions are examples. +------------------------------------------------------------ | field +------------------------------------------------------------ A field is a commutative ring (R,+,*,0,1) such that (R,+,0) and (R setminus 0,*,1) are both commutative groups. +------------------------------------------------------------ | theorem of Zorn +------------------------------------------------------------ By a theorem of Zorn (1933), every alternative, quadratic, real non-associative algebra without zero divisors is isomorphic to the 8-dimensional octonions O. +------------------------------------------------------------ | Theorem of Hurwitz +------------------------------------------------------------ Theorem of Hurwitz: the normed composition algebras with unit are: real numbers, complex numbers, quaternions; and octonions. +------------------------------------------------------------ | Theorem of Kervaire and Milnor +------------------------------------------------------------ Theorem of Kervaire and Milnor In 1958, Kervaire and Milnor proved independently of each other that the finite-dimensional real division algebras have dimensions 1,2,4, or 8. +------------------------------------------------------------ | Theorem of Adams +------------------------------------------------------------ Theorem of Adams In 1960, Adams proved that a continuous multiplication in R^n+1 with two-sided unit and with norm product exists only for n+1 = 1,2,4, or 8. +------------------------------------------------------------ | Theorem of Hurwitz +------------------------------------------------------------ Theorem of Hurwitz: the normed composition algebras with unit are: real numbers complex numbers quaternions octonions +------------------------------------------------------------ | Theorems of Sylov +------------------------------------------------------------ Theorems of Sylov If G is a finite group of order |G|=p^n q, where p is a prime number, then G has a subgroup of order p^n. Such groups are called Sylov groups and all of them are isomorphic. Furthermore, the number N of different p-Sylov groups in G satisfies N =1 mod (p). This file is part of the Sofia project sponsored by the Provost's fund for teaching and learning at Harvard university. There are 39 entries in this file. COUNT: 39