Exercise 2.3.7 (Subgroups of $Z_{48}$)

Question

Let $Z_{48} = \langle x \rangle$ and use the isomorphism $\mathbb{Z}/48\mathbb{Z} \simeq Z_{48}$ given by $\overline{1} \mapsto x$ to list all subgroups of $Z_{48}$ as computed in the preceding exercise.

Richard Ganaye · Accepted Answer

\begin{proof} 
In multiplicative notations, the lattice of the subgroups of $Z_{48}$ is
\begin{center}
\begin{tikzpicture}

ode (48) at (0,0) {$\qquad \langle x^{48} angle = \langle 1 angle$} ;

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

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

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

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

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

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

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

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

ode (1) at (-6,6) {$\qquad \qquad \langle x^{1} angle = Z_{48}$} ;
\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{proof}

Exercise 2.3.7 (Subgroups of $Z_{48}$)

Answers

Comments

Exercise 2.3.7 (Subgroups of $Z_{48}$)

Answers

Comments

Add answer