Q:

Which of the following groups are isomorphic (TIFR 2014)?

Which of the following groups are isomorphic?
(a) $\mathbb{R}$ and $\mathbb{C}$
(b) $\mathbb{R}^*$ and $\mathbb{C}^*$
(c) $S_3\times \mathbb{Z}_4$ and $S_4$
(d) $\mathbb{Z}_2\times \mathbb{Z}_2$ and $\mathbb{Z}_4$ 
Here option (d) is not correct because one is not cyclic and other one is cyclic. Also $\mathbb{R}$ and $\mathbb{C}$ are vector space isomorphic (over the field $\mathbb{Q}$). But i cannot conclude correct answer help me!

A:

(a) As you have observed, these are isomorphic as $\mathbb{Q}$-vector spaces. A fortiori their underlying additive groups are isomorphic.
(b) Count the solutions of $x^4=1$ in $\mathbb{R}^*$ resp. $\mathbb{C}^*$. (Alternatively, look at the map $x \mapsto x^2$ and check for surjectivity.)
(c) The abelianization of $S_3 \times \mathbb{Z}/4$ is $\mathbb{Z}/2 \times \mathbb{Z}/4$, but the abelianization of $S_4$ is $\mathbb{Z}/2$. (Alternatively, notice that there is no epimorphism $S_4 \to \mathbb{Z}/4$ since the kernel would contain the commutator subgroup $A_4$, but it would have only $4!/4=6$ elements.)
(d) You are right, $\mathbb{Z}/2 \times \mathbb{Z}/2$ is not cyclic.

