# Groups without commuting elements (TIFR 2015 problem 4)

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, $|S|=1$.