Let $n$ be a positive integer such that every group of order $n$ is cyclic. Show the following. (A) For all prime numbers $p$, $p^2$ does not divide $n$. (B) If $p$ and $q$ are prime divisors of $n$, then $p$ does not divide $q - 1$. (Hint: Consider $2 \times 2$ matrices $$\left[\begin{array}{ll} x & y \\ 0 & 1 \end{array}\right]$$ with $x, y \in \mathbb{Z}/q\mathbb{Z}$ and $x^p = 1$.) (C) Show that $(n, \phi(n)) = 1$, where $\phi(n)$ is the number of integers $m$ such that $1 \leq m \leq n$ with $\gcd(n, m) = 1$.
Let $n$ be a positive integer such that every group of order $n$ is cyclic. Show the following.\\
(A) For all prime numbers $p$, $p^2$ does not divide $n$.\\
(B) If $p$ and $q$ are prime divisors of $n$, then $p$ does not divide $q - 1$. (Hint: Consider $2 \times 2$ matrices
$$\left[\begin{array}{ll} x & y \\ 0 & 1 \end{array}\right]$$
with $x, y \in \mathbb{Z}/q\mathbb{Z}$ and $x^p = 1$.)\\
(C) Show that $(n, \phi(n)) = 1$, where $\phi(n)$ is the number of integers $m$ such that $1 \leq m \leq n$ with $\gcd(n, m) = 1$.