Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then G = 
ii) Ifx and y are elements of a non-abelian group  such that   then where is the identity of G with respect to 
iii) There exists a unique non-abelian group of prime order.
iv) If  where A is a group, then 
v) If H and K are normal subgroups of a group G, then 

i) If a group \( G \) is isomorphic to one of its proper subgroups, then \( G = \mathbb{Z} \).

This statement is false. For example, consider the group of integers under addition, \( (\mathbb{Z}, +) \). \( (\mathbb{Z}, +) \) is isomorphic to its proper subgroup \( (2\mathbb{Z}, +) \), which consists of even integers. However, \( \mathbb{Z} \) is not equal to \( \mathbb{Z} \), so the statement is false.

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