Question 12
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 cardinal of $H$. Prove that
a) $|H \cap xH| \geq 2p - n$, for any $x \in G$, where $xH = \{xh : h \in H\}$.
b) If $p \gt \frac{3n}{4}$, then $G$ is commutative.
c) If $\frac{n}{2} \lt p \leq \frac{3n}{4}$, then $G$ is non-commutative.
$$
H = \{x : x \in G \text{ and } x^2 = e\},
$$
where $e$ is the neutral element of $G$. Let $p = |H|$ be the cardinal of $H$. Prove that
a) $|H \cap xH| \geq 2p - n$, for any $x \in G$, where $xH = \{xh : h \in H\}$.
b) If $p \gt \frac{3n}{4}$, then $G$ is commutative.
c) If $\frac{n}{2} \lt p \leq \frac{3n}{4}$, then $G$ is non-commutative.