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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 )
(\mathbb{R}) and (\mathbb{C}) are isomorphic:
Both these spaces are vector spaces ove (\mathbb{Q}).
We state a result without proof.
Theorem: Let (V) be an infinite dimensional vector space over a countable field (F). Then the dimension of (V) over (F) is (|V|) (the cardinality of (V) ).
This immediately tells us that (\mathbb{R}) has dimension (|R| = 2^{\aleph _0 } ).
Now the dimension of (\mathbb{C}) over (\mathbb{Q}) is (dim(\mathbb{R}^2) = 2 \times 2^{\aleph _0 } =2^{\aleph _0 } ).
What did we just show? We showed that (dim(\mathbb{C})=dim(\mathbb{R}) ) over (\mathbb{Q}).
Therefore they are isomorphic as vector spaces. So they are isomorphic as groups with respect to addition.
(\mathbb{R}^* ) and (\mathbb{C}^*) are not isomorphic:
(\mathbb{C}^) has an element of order 4, namely (i) has order 4. If there was an isomorphism then the corresponding element in (\mathbb{R}^) will also have order 4. But there is no element of order 4 in (\mathbb{R}^*). Hence we conclude that these two groups are not isomorphic.
(S_3 \times \mathbb{Z}_4) and (S_4) are not isomorphic:
The order of (((1 2 3), [1] ) \in S_3 \times \mathbb{Z}_4) is 12. (S_4) does not contain an element of order 12.
(\mathbb{Z}_2 \times \mathbb{Z}_2 ) and ( \mathbb{Z}_4 ) are not isomorphic:
(\mathbb{Z}_2 \times \mathbb{Z}_2 ) is not cyclic and ( \mathbb{Z}_4 ) is cyclic. So they can not be isomorphic.
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