Groups Notes | Study GATE Computer Science Engineering(CSE) 2023 Mock Test Series - GATE

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 .