Classical

Algebra

 

 

 

 

Questions Answers
 

Let S= {0,1,2,3,4,5.6.7} and  ⊗ denotes multiplication modulo 8 , i.e. x ⊗ y= (xy) mod 8

(i) Prove that ({0,1}, ⊗ } is not a group
(ii) Write 3 distinct groups (G,⊗) where G S and  G has 2 elements

 

Solution 
(i) ⊗ does not have an inverse
(ii) {1,3}, {1,5}, {1,7}


Let G1 and G2 be any two subgroups of G.
(i) Show that G1 G2 is also a subgroup of G
(ii) Is G1 G2  always a subgroup of G ?


Solution 
Let G1 and G2 be any two subgroups of G. Then  G1G2  ≠Φ , since at least identity element e is common to both G1  and G2 . the order to prove that G1 ∩ G2 is  subgroup it is sufficient to prove that

a∈G1  ∩ G2 , b ∈G1 ∩ G2

⇒ ab-1∈G1 ∩ G2

Now, a∈G1 ∩ G2,    a ∈G1 and a ∈G2

 b∈G1 ∩ G2,    b ∈G1 and b∈G2

But G1, G2 are subgroups , therefore 
a ∈G1, b∈G1

⇒ ab-1∈ G1, a∈G2, b ∈G2

⇒ ab-1 ∈ G2
Finally ab-1 ∈ G1, ab-1 ∈ G2

⇒ ab-1 ∈ G1 ∩ G2

Thus we have shown that a ∈ G1 ∩ G2 , b ∈ G1 ∩ G2 

⇒ ab-1 ∈ G1 ∩G2

Hence G1 ∩G2 is a subgroup of G

(ii) Solution : No

It is not necessarily that G1 ∪ G2 is a subgroup of G .
A group (G,*) must satisfy four conditions :
         closure,associtivity, identity and inverse
Since G1 and G2 are subgroups , ie groups themselves, G1 
∩ G2 also satisfy closure,associative and identity.
Since every element has its unique  G
∩ G2 is a subgroup ie if a has a-1 in G , then a-1 is also a inverse of a in G1 and G2
a-1  
∈ G1 ∩ G2




Let (A,*) be a semigroup. Furthermore,for every a and b in A, if a  ≠ b, then a* b  ≠ b*a
(i)  Show that for every a in A , a*a=a
(ii) Show that for every a,b in A , a* b*a =a
(iii) Show that for every a,b,c in A , a*b*c = a*c


Solution
(i) A is a semigroup

∴ (a*b) * c = a*(b*c)
Putting, b=a  and c=a , we get
      (a*a)*a = a * (a*a) and A is not abelian
     a*a =a

⇒ b*b=b
(ii) Let b 
∈ A , then 
Premultiplying by a, we get 
                  a*(b*b) =a*b
                  (a*b)*b=(a*b)
 
⇒               a*b = a
 ⇒               a*b*a = (a*b)*a

 

Using Equation (ii) , a*b*a=a*a
            =a 
(III) a*b*c=(a*b)*c
             = a*c

 

Let ({p*q},*) be a semigroup where p*p =q. Show that
(i) p*q=q*p
(ii) q*q=q


Solution
Let S = ({p,q},*) be a semi group
Since S is a semi group, it is closed and associative, ie
a) a*b 
∈ S a v b ∈ S
b) a*(b*c)=(a*b)*c
∀ a,b,c ∈ S
(i)                  p*q = p*(p*p
)
                           = (p*p)*p
                          = q*p
(ii)  to prove q*q =q
Since , S is closed p*q 
∈ S = {p,q}
therefore  p*q =p, or p*q=q
Case I - Let p*q =p
               q*q =q*(p*p)
                    = (q*p)*p
                   = (p*q)*p
                   =p*p
                 = q
Case II: Let p*q=q
 
                
 q*q =(p*p)*q
  
                    = p*(p*q)
  
                    = (p*q)
   
                  =p*q 
     
                 = q