9 The set \(S\) consists of the numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\). ( \(\mathbb { Z }\) denotes the set of integers \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)
- Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
- Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
(a) The numbers \(3 ^ { 2 n }\), where \(n \in \mathbb { Z }\).
(b) The numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\) and \(n \geqslant 0\).
(c) The numbers \(3 ^ { \left( \pm n ^ { 2 } \right) }\), where \(n \in \mathbb { Z }\).
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