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,*), then (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.
For the proof of the uniqueness of the identity element and the uniqueness of the inverse of an element in a GrouP, see ElementaryGroupTheory.
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. We usually only use additive notation when the group is commutative, ie a+b=b+a for all elements.
Some important concepts: SubGroups, PermutationGroup?s. Some examples of groups with additional structure are RinGs, ModulEs and LieGroup?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.
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(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 and FielDs.