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

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'''MathematicalGrouP - definition''' 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. This turns out to be unique. 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. This turns out to be unique. Usually the operation, whatever it ''really'' is, is denoted like multiplication; we write a*b for the ''product'' of a and b, 1 for the identity and a^-1 for the inverse of a. Otherwise, the operation is denoted like addition; we write a+b for the sum of a and b, 0 for the identity and -a for the inverse of a. Some important concepts: SubGroup''s'', PermutationGroup''s''. Some examples of groups with additional structure are RinG''s'' and ModulE''s''. ---- 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 RinG''s'', IdeaL''s'' and FielD''s''.