Given below are two statements :

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

Statement (II): There is no homomorphism existing from (Z,+) to (Q,+); where Z is a set of integers and Q is a set of rational numbers. In the light of the above statements, choose the most appropriate answer from the options given below:

1. ABoth Statement I and Statement II are correct
2. BBoth Statement I and Statement II are incorrect
3. CStatement I is correct but Statement II is incorrect
4. DStatement I is incorrect but Statement II is correct

Pick an option to check yourself, or open the answer below.

Show answer & explanation

Correct answer: C. Statement I is correct but Statement II is incorrect

Explanation

A non-empty finite subset closed under the group operation is a subgroup (Statement I true). There are homomorphisms from (Z, +) to (Q, +), for example the inclusion n ↦ n or the zero map, so Statement II is false.