F3 - groups of units modulo n

U_{k} (n) = {x \in U (n) ∣ x mod k = 1}

U (n)

as an external direct product

\begin{matrix} U (s t) ≅ U (s) \oplus U (t) \\ U_{s} (s t) ≅ U (t) \\ U_{t} (s t) ≅ U (s) \end{matrix}

corollary

let $m = n_{1} n_{2} \dots n_{k}$ , where $gcd (n_{i}, n_{j)} = 1$ for $i \neq j$ , then:

U (m) ≅ \oplus_{i = 1}^{k} U (n_{i})

\begin{aligned} U (105) & ≅ U (7) \oplus U (15) \\ ≅ U (21) \oplus U (5) \\ ≅ U (3) \oplus U (5) \oplus U (7) \end{aligned}

\begin{matrix} U (7) ≅ U_{15} (105) \\ U (15) ≅ U_{7} (105) \\ U (21) ≅ U_{5} (105) \\ ⋮ \end{matrix}

the condition $gcd (s, t) = 1$ is necessary as $| U (20) | = 8$ and $| U (10) \oplus U (2) | = 4$
$Z_{n}$ have the simplest structures and are the easiest to compute with
the external direct products of groups of the form $Z_{n}$ are only slightly more complicated in structure and computability
therefore, describing a finite abelian group as such a direct product is useful
any $U$ - group can be expressed an an external direct product of cyclic groups using the corollary and the facts:

\begin{matrix} U (2) ≅ {0} \\ U (4) ≅ Z_{2} \\ U (2^{n}) ≅ Z_{2^{n - 2}} \oplus Z_{2} for n \geq 3 \\ U (p^{n}) ≅ Z_{p^{n} - p^{n - 1}} for p an odd prime \end{matrix}

\begin{aligned} U (105) & = U (3 \cdot 5 \cdot 7) \\ ≅ U (3) \oplus U (5) \oplus U (7) \\ ≅ Z_{2} \oplus Z_{4} \oplus Z_{6} \end{aligned}

\begin{aligned} U (144) & = U (16) \oplus U (9) \\ ≅ Z_{4} \oplus Z_{2} \oplus Z_{6} \end{aligned}

\begin{aligned} Aut (Aut (Aut (Z_{27}))) & ≅ Aut (Aut (U (27))) \\ ≅ Aut (Aut (Z_{18})) & ≅ Aut (U_{18}) \\ ≅ Aut (Z_{6}) & ≅ U (6) ≅ Z_{2} \end{aligned}