Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Lemma

Lemma for Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo $n$

There are only $5$ numbers $n$ with the property that $\map \phi n = 8$, and they are $15$, $16$, $20$, $24$ and $30$.

Proof

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Let $p$ be a prime factor of $n$.

By Euler Phi Function is Multiplicative:

$p - 1 = \map \phi p \divides \map \phi n = 8$

so $p \in \set {2, 3, 5}$.

Let $i, j, k \in \Z^{\ge 0}$ such that $n = 2^i 3^j 5^k$.

By Euler Phi Function is Multiplicative:

$8 = \map \phi n = \map \phi {2^i} \map \phi {3^j} \map \phi {5^k}$

has only $5$ solutions:

$\paren{i, j, k} = \paren{0, 1, 1}, \paren{4, 0, 0}, \paren{2, 0, 1}, \paren{3, 1, 0}$ and $\paren{1, 1, 1}$.

$\blacksquare$