Given below are two statements : Statement (I) : If H is non empty finite…

2023

Given below are two statements :

Statement (I) : If H is non empty finite subset of a group G and ab∈H ∀ a, b∈H, then H is also a group

Statement (II) : There is no homomorphism exist from (Z, +) to (Q, +); where Z is set of integers and Q is set of rational number.

In the light of the above statements, choose the most appropriate answer from the options given below :

  1. A.

    Both Statement I and Statement II are correct

  2. B.

    Both Statement I and Statement II are incorrect

  3. C.

    Statement I is correct but statement II are incorrect

  4. D.

    Statement I is incorrect but statement II are correct

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Correct answer: C

Answer: Statement (I) is true and Statement (II) is false. Therefore the correct choice is the one stating that Statement (I) is correct but Statement (II) is incorrect.

Proof of Statement (I): Let H be a nonempty finite subset of a group G with the property that the product of any two elements of H lies in H (closure). Pick any element a in H. Consider the sequence a, a^2, a^3, ... Since H is finite, two terms must be equal, say a^m = a^n with n>m. Then a^{n-m} = e (the identity of G), so e ∈ H. Now for a, if a^k = e then a^{k-1} is the inverse of a and lies in H. Thus every element of H has an inverse in H, and H contains the identity and is closed under the operation, so H is a subgroup of G.

Explanation of Statement (II): The statement claiming there is no homomorphism from (Z,+) to (Q,+) is false. Any additive group homomorphism φ: Z → Q is determined by φ(1), which can be any rational number q. Then φ(n) = n·q for all integers n defines a homomorphism. For example, choosing q = 1/2 gives the nontrivial homomorphism φ(n) = n/2. The zero map (q = 0) is another example. Hence homomorphisms do exist.

  • Key point for Statement (I): finiteness + closure ⇒ identity and inverses lie in H.

  • Key point for Statement (II): homomorphisms Z → Q correspond to choosing any rational image for 1, giving φ(n)=n·q.

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