Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

<-- Previous | Newer --> | Current: 980265271 RoseParks at Tue, 23 Jan 2001 15:54:31 +0000.

MathematicalGrouP The concept of a Group is one of the foundations of ModernAlgebra. Its definition is brief. A Group is a NonEmpty SeT, say G and a BinaryOperation, say, "*" denoted (G,*) such that: 1) G has CloSure, that is, if a and b belong to G, then a*b belongs to G. 2) The operation * is associative, that is, if a, b, and c belong to G, (a*b)*c=a*(b*c). 3) G contains an identity element, say e, that is, if a belongs to G then e*a = a and a*e = a. 4) Every element in G has an inverse, that is, if a belongs to G, there is an element b in G such that a*b=e=b*a. Sets of BiJections form groups under composition, providing they are closed under this operation and inverses. A HomoMorphism from a group to a such a set is called a representation thereof, and it is called faithful if different group elements correspond to different BiJections. CayleysTheorem states that every group has such a representation, so the study of groups can be considered as the study of permutations. A subset of G closed under * forms a SubGroup in an obvious way. Every group contains at least two subgroups, itself and the trivial group containing only the identity. Every other element is contained in a minimal subgroup isomorphic either to the IntegerNumbers or some ModularArithmetic, treated as groups under addition. The KerneL of a group HomoMorphism consists of those elements that get mapped to an identity. As it turns out, the possible KerneLs of a group are precisely the NormalSubGroups - those subgroups H such that g*H=H*g for all g in G - and this fixes the homomorphic image of the group up to isomorphism. ---- A Group that we are introduced to in elementary school is the integers under addition. Thus, let Z be the set of integers={?-4,-3,-2,-1,0,1,2,3,4?} and let the symbol "+" indicate the operation of addition. Then, (Z,+) is a GrouP. Proof: 1) If a and b are integers then a+b is an integer: Closure. 2) If a, b, and c are integers, then (a+b)+c=a+(b+c). Associativity. 3) 0 is an integer and for any integer a, a+0=a. (Z,+) has an identity element. 4) If a is an integer, then there is an integer b= (-a), such that a+b=0. Every element of (Z,+) has an inverse. ---- Question: Given the set of integers, Z, as above, and the operation multiplication, denoted by "x" is (Z,x) a Group? 1) If a and b are integers then axb is an integer. Closure. 2) If a, b, and c are integers, then (axb)xc=ax(bxc). Associativity. 3) 1 is an integer and for any integer a, ax1=a. (Z,x) has an identity element. 4) '''BUT''', if a is an integer, there is not necessarily an integer b =1/a such that (a)x(1/a)=1. Then, every element of (Z,x) does not have an inverse. For example, given the integer 4, there is no integer b such that 4xb=1. Therefore, (Z,x) is '''not''' a Group. It is a weaker type of object sometimes called a SemiGroup. ---- '''Question''': Given the set of rational numbers Q, that is the set of number a/b such that a and b are integers, but b is not = to 0, and the operation multiplication, denoted by "x," is (Q,x) a GrouP?---- Groups are important, too, because they are a fundamental algebraic structure. We can investigate groups with added properties like CommutativeGroups and NonCommutativeGroups, and also build other structures based on the notion of a group, but with more than one operations and more properties. This brings us to RinGs, IdeaLs and FielDs.