Exercise 2.5.9 (Lattices of subgroups of $ \mathbb{Z}/16 \mathbb{Z}$, $ \mathbb{Z}/24 \mathbb{Z}$, $ \mathbb{Z}/48 \mathbb{Z}$)

Question

Draw the lattices of subgroups of the following groups:
$$	ext{\bf(a)}\ \mathbb{Z}/16 \mathbb{Z} \qquad	ext{\bf(b)}\ \mathbb{Z}/24 \mathbb{Z} \qquad 	ext{\bf(c)}\ \mathbb{Z}/48 \mathbb{Z}.$$

Richard Ganaye · Accepted Answer

\begin{proof} 
Already done in Exercise 2.3.6. for $ \mathbb{Z}/48 \mathbb{Z}$.
\begin{enumerate}
\item[{\bf(a)}] Lattice of subgroups of $\mathbb{Z}/16 \mathbb{Z}$:

\bigskip
\begin{center}
\begin{tikzpicture}

ode (16)   at (0,0) {$\qquad \ \ \langle \overline{16}angle = \langle 0 angle$};

ode (8)   at (0,1) {$\langle \overline{8}angle$};

ode (4)   at (0,2) {$\langle \overline{4}angle$};

ode (2)   at (0,3) {$\langle \overline{2}angle$};

ode (1)   at (0,4) {$\qquad \qquad  \langle \overline{1}angle = \mathbb{Z}/16 \mathbb{Z}$};
\draw (16) edge  (8)  ;
\draw (8) edge  (4)  ;
\draw (4) edge  (2)  ;
\draw (2) edge  (1)  ;
\end{tikzpicture}
\end{center}

\item[{\bf(b)}] Lattice of subgroups of $\mathbb{Z}/ 24 \mathbb{Z}$:

\bigskip
\begin{center}
\begin{tikzpicture}

ode (24) at (-2, 1) {$\qquad\langle \overline{24} angle = \langle \overline{0} angle$} ;

ode (12) at (-4, 2) {$\langle \overline{12}angle$} ;

ode (6)   at (-6, 3) {$\langle \overline{6}angle$};

ode (3)   at (-8, 4) {$\langle \overline{3}angle$};

ode (8) at (0,3) {$\langle \overline{8} angle$} ;

ode (4) at (-2,4) {$\langle \overline{4} angle$} ;

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

ode (1) at (-6,6) {$\qquad \qquad \langle \overline{1}angle = \mathbb{Z}/24\, \mathbb{Z}$} ;
\draw (1) edge  (2)  ;
\draw (2) edge  (4)  ;
\draw (4) edge  (8)  ;
\draw (3) edge  (6)  ;
\draw (6) edge  (12)  ;
\draw (12) edge  (24)  ;
\draw (1) edge  (3)  ;
\draw (2) edge  (6)  ;
\draw (4) edge  (12)  ;
\draw (8) edge  (24)  ;
\end{tikzpicture}
\end{center}
\item[{\bf(c)}] Lattice of subgroups of $\mathbb{Z}/ 48 \mathbb{Z}$:

\bigskip
\begin{center}
\begin{tikzpicture}

ode (48) at (0,0) {$\qquad \langle\overline{48} angle = \langle \overline{0} angle$} ;

ode (24) at (-2, 1) {$\langle \overline{24} angle$} ;

ode (12) at (-4, 2) {$\langle \overline{12}angle$} ;

ode (6)   at (-6, 3) {$\langle \overline{6}angle$};

ode (3)   at (-8, 4) {$\langle \overline{3}angle$};

ode (16) at (2,2) {$\langle \overline{16} angle$} ;

ode (8) at (0,3) {$\langle \overline{8} angle$} ;

ode (4) at (-2,4) {$\langle \overline{4} angle$} ;

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

ode (1) at (-6,6) {$\qquad \qquad \langle \overline{1}angle = \mathbb{Z}/48\, \mathbb{Z}$} ;
\draw (1) edge  (2)  ;
\draw (2) edge  (4)  ;
\draw (4) edge  (8)  ;
\draw (8) edge  (16)  ;
\draw (3) edge  (6)  ;
\draw (6) edge  (12)  ;
\draw (12) edge  (24)  ;
\draw (24) edge  (48)  ;
\draw (1) edge  (3)  ;
\draw (2) edge  (6)  ;
\draw (4) edge  (12)  ;
\draw (8) edge  (24)  ;
\draw (16) edge  (48)  ;
\end{tikzpicture}
\end{center}

\end{enumerate}
\end{proof}

Exercise 2.5.9 (Lattices of subgroups of $ \mathbb{Z}/16 \mathbb{Z}$, $ \mathbb{Z}/24 \mathbb{Z}$, $ \mathbb{Z}/48 \mathbb{Z}$)

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Exercise 2.5.9 (Lattices of subgroups of $ \mathbb{Z}/16 \mathbb{Z}$, $ \mathbb{Z}/24 \mathbb{Z}$, $ \mathbb{Z}/48 \mathbb{Z}$)

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