Exercise 2.3.1 (Subgroups of $Z_{45}$)

Question

Find all subgroups of $Z_{45} = \langle x \rangle$, giving a generator for each. Describe the containments between these subgroups.

Richard Ganaye · Accepted Answer

\begin{proof} 
The positive divisors of $n = 45 = 3^2\cdot 5$ are $1, 3, 3^2 = 9, 5, 3\cdot 5 = 15, 3^2\cdot 5 = 45$, so the set of positive divisors is
$$D_{45} = \{1,3 ,9, 5, 15, 45\} .$$
By Theorem $7$, for each $a \in D_{45}$, there is a unique subgroup  $\langle x^d angle \leq Z_{45}$  where $d = n/a$. Moreover 
$$\langle x^{n/a} angle \leq \langle x^{n/b} angle \iff a \mid b,$$
or equivalently, for divisors $e, f$ of $45$,
$$\langle x^{e} angle \leq \langle x^{f} angle \iff f \mid e,$$
This gives the following diagram of inclusions (inclusions go from bottom to top):
\bigskip
\begin{center}
\begin{tikzpicture}

ode (45) at (0,0) {$ \qquad \langle x^{45} angle = \langle 1 angle$} ;

ode (15) at (-2, 1) {$\langle x^{15} angle$} ;

ode (5) at (-4, 2) {$\langle x^5 angle$} ;

ode (9) at (2,2) {$\langle x^{9 }angle$} ;

ode (3) at (0,3) {$\langle x^{3} angle$} ;

ode (1) at (-2,4) {$\qquad \langle x^1 angle = Z_{45}$} ;
\draw (45) edge  (9)  ;
\draw (9) edge  (3)  ;
\draw (3) edge  (15)  ;
\draw (3) edge  (1)  ;
\draw (5) edge  (15)  ;
\draw (15) edge  (45)  ;
\draw (1) edge  (5)  ;
\end{tikzpicture}
\end{center}
\end{proof}

Exercise 2.3.1 (Subgroups of $Z_{45}$)

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Exercise 2.3.1 (Subgroups of $Z_{45}$)

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