Which of the following statements are true? Give reasons for your answ

Question

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 $G = \mathbb{Z}$ .
ii) If x and y are elements of a non-abelian group (G, ) such that $x y = y x$ , then $x = e$ or $y = e$ , where e is the identity of G with respect to $$.
iii) There exists a unique non-abelian group of prime order.
iv) If $(a, b) \in A \times A$ , where A is a group, then $o((a, b)) = o(a)o(b)$ .
v) If H and K are normal subgroups of a group G, then $hk = kh \quad \forall \quad h \in H, k \in K$ .

10 Jan 2026

Answer :

Word Count : 215

i) False. If a group (G) is isomorphic to one of its proper subgroups, it does not necessarily imply any specific equation unless further context is given. For finite groups, this is impossible because the order of a proper subgroup is strictly less than the order of the group, so a finite group _________ _________ ________ _____ __________ _____ ______ ______ __________ _________.
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