RO-NMO 2010 Problem 123

Let $G$ be a finite group of order $n$ . Define the set $$H=\{x:x\in G\text{ and }x^2=e\},$$ where $e$ is the neutral element of $G$ . Let $p=|H|$ be the cardinality of $H$ . Prove that a) $|H\cap xH|\ge 2p-n$ , for any $x\in G$ , where $xH=\{xh:h\in H\}$ . b) If $p>\frac{3n}{4}$ , then $G$ is commutative. c) If $\frac{n}{2}<p\le\frac{3n}{4}$ , then $G$ is non-commutative.

Marian Andronache