Question 19
Suppose $(G, \cdot)$ is a finite group with unity $e$, $a$ is an element in $G \setminus \{e\}$ and $p$ is a prime number such that $x^{p+1} = a^{-1} x a$, for all $x \in G$.
a) Show that there is $k \in \mathbb{N}^*$ such that $\text{ord}(G) = p^k$.
b) Prove that $H = \{x \in G \mid x^p = e\}$ is a subgroup of $G$ and
$$
(\text{ord}(H))^2 \gt \text{ord}(G).
$$
a) Show that there is $k \in \mathbb{N}^*$ such that $\text{ord}(G) = p^k$.
b) Prove that $H = \{x \in G \mid x^p = e\}$ is a subgroup of $G$ and
$$
(\text{ord}(H))^2 \gt \text{ord}(G).
$$