Question:

Let \(S\) be the collection of isomorphism classes of groups \(G\) such that every element of G commutes with only the identity element and itself. Then what is \(|S|\)?

Discussion:

Given any \(g\in G\), it commutes with few obvious elements: \(e,g,g^2,…\) , i.e, all integral powers of \(g\).

So by given condition, this whole set {\(e,g,g^2,…\)} is same as the set {\(e,g\)}. So any element in \(G\) must have order \( \le 2\).

Now let us look at \(e\). The identity commutes with every element. But by given condition, \(e\) commutes with \(e\) only. That implies there is no other element in \(G\).

So, \(G=(e)\).

So, \(|S|=1\).