VIRTUAL GATE TEST SERIES

Question

Comments

@MiNiPanda

here inverse not exists

So, both will not a group D)

rt?

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Oh! I thought "modulo" is also present in (1).

Yes then answer is D.

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answer is given A.

Answer 1

(i) Zn has element { 0 , 1 , 2 , . . . , n-1 }

We can prove that this set is Group under the Modulo Arithmatic

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(ii) Zn* has element { 1 , 2 , 3 , 4 , … , n-1 }

For this :

(1) closure property satisfied because after doing multiplication we do modulo operation so at last we fall in one of the element from the set

(2) Multiplication with modulo is associative

(3) identity element is 1

(4) Inverse : This is interesting case => There is theorem that says that Multiplicative inverse exist for element iff element is relative prime to the n (here n is Zn )

and we are given n is prime so for any element a ,GCD(n,a) = 1 so for any element in our set we have multiplicative inverse

So inverse exist for every element

So (ii) is also group

 

Option A is correct

Answer 2

For group 1, the group elements are {1, 2, 3, 4…/+}

For group 2, the elements are   { for prime number is 2 then all even numbers  2, 4, 6, ….

                                                    for prime number is 3 then all multiples of 3 like 3, 6, 9, 12…

                                                    for prime number 5, its multiples  5, 10, 15, 20and so on for all primes / *}

Both will form groups, hence option A is correct