Group theory

A set (G,o) is called group if it is satisfy 4 condition
G1--closure -
       a is in G b is in G then =>aob is in G, for all a,b is in G.
G2-- associate--
       For all a,b,c is in G
       (aob)oc=ao(boc).
G3-- identity--
       For all a is in G  there are exist e in G s.t.
       aoe=a. ,for all a in G.
G4-- inverse--
         b is called inverse of a if
         aob=e for all a,b is in G


Then (G,o) is called a group....CLICK HERE

Abelian Group— A group G is called abelian group if it is satisfy a condition

G5- commutative-

                        aob = boa  ∀a,b∈G……..

Groupoid or binary Algebra—

A non-empty set G with a operation “o” is called Groupoid if its satisfy G1 . that is closure ……….

Semi Group— A binary structure (G,o) is called semi group is its satisfy two condition G1 and G2 that is closure and associate………

Monoid-A semi group is called monoid if there are exist “1” ∈ G such that 1om=mo1 =m ∀m∈G

Finite and infinite GroupàA Group is called finite group if it have finite number of elements. And it is called infinite group if it have infinite number of element.

Order of a groupè The number of element in finite Group is called order of a group. And its denoted by o(G).

Qà show that set of integer (I,+)  is a group.

Solveà

We know that the set of  integer I=(……..-3,-2,-1,0,1,2,3……………)

We have to show – set of integer is a group.

For this we will prove four exioms.

G1à closure :-  for any integer a,b ∈I

                        a,b∈I=> a+b∈I  for all a,b∈I

G2à associative:- for a,b,c ∈I

                        a+(b+c)=(a+b)+c , for all a,b,c∈I

G3à Identity:- if a∈I then there are exist 0 in I st

                        a+0=a for all a in I

G4à Inverse:- if a∈I then there are exist –a in I such that

                             a+(-a)=0 for all a∈I

I is satisfies all condition there for (I,+) is a group.

*But set of natural number (N,+) is not a group.

* set of natural number (N, .) is not a group.

*set of real number (R,+) and (R,. ) are group.

* A set (G,*) will be group defined as a*b=a+b+1 for all a,b in G.

*set of natural number is not a group with respect to addition i.e. (N,+).
solve:- let I = {1,2,3....} is a set of Natural number .
to prove :- (N,+) not a gruop.
for this , 
1. closure:-  a,b∈N=> a+b∈N  for all a,b∈N.
                    for ex. 1,3∈N =>  4∈N 
2.Associate :- a,b,c ∈N  then (a.b).c =a.(b.c) for all a,b,c∈N 
                    for ex. (1.2).3=1.(2.3)
                                 2.3   = 1.6
                                   6      =6
therefore it satisfies associate law
3. Identity :- for all a∈N there are not exist 0 in N st a+0=a  i.e. it is no satisfies identity law and hence it is not a gruop. CLICK HERE

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