Groups Computer Science Engineering (CSE) Notes | EduRev
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Computer Science Engineering (CSE) : Groups Computer Science Engineering (CSE) Notes | EduRev
Groups|Discrete Mathematics
Algebraic Structure
A non empty set S is called an algebraic structure w.r.t binary operation (*) if (a*b) belongs to S for all (a*b) belongs to S, i.e (*) is closure operation on ‘S’.
Ex : S = {1,-1} is algebraic structure under *
As 1*1 = 1, 1*-1 = -1, -1*-1 = 1 all results belongs to S.
But above is not algebraic structure under + as 1+(-1) = 0 not belongs to S.
Semi Group
An algebraic structure (S,*) is called a semigroup if a*(b*c)=(a*b)*c for all a,b,c belongs to S or elements follow associative property under * .
Ex : (Set of integers, +) is semigroup.
But (Matrix ,*) is semigroup.
Monoid
A Semigroup (S,*) is called a monoid if there exists an element e in S such that (a*e) = (e*a) = a for all a in S. This element is called identity element of S w.r.t *.
Ex : (Set of integers,*) is Monoid as 1 is an integer which is also identity element . (Set of natural numbers, +) is not Monoid as there doesn’t exists any identity element. But this is Semigroup.
Group
A monoid (S,*) is called Group if to each element there exists an element b such that (a*b) = (b*a) = e . Here e is called identity element an b is called inverse of the corresponding element. (Set of rational number , *) is not Group because there doesn’t exists inverse for 0 Thus for a Group:
It should be
1) Algebric Structure
2) Semigroup
3) Moniod
4) have inverse.
Abelian Group
A group (G,*) is said to be abelian if (a*b)=(b*a) for all a,b belongs to G. Thus Commutative property should hold .
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