Exercise 2.5.1 (Sublattices with $H$ and $K$)

Question

Let $H$ and $K$ be subgroups of $G$. Exhibit all possible sublattices which show only $G$, $1$, $H$ and $K$ and their joins and intersections. What distinguish the different drawings?

Richard Ganaye · Accepted Answer

\begin{proof} 
We assume that  $H$ and $ K$ are nontrivial distinct proper subgroups. Then the wanted sublattices are
\bigskip
\begin{center}
\begin{tikzpicture}

ode (1) at (0,0) {$1$} ;

ode (HcapK) at (0,1) {$H \cap K$} ;

ode (H) at (-1,2) {$H$} ;

ode (K) at (1,2) {$K$} ;

ode (HK) at (0,3) {$\langle H, K angle$} ;

ode (G) at (0,4) {$G$} ;
\draw (1) edge  (HcapK)  ;
\draw (HcapK) edge  (H)  ;
\draw (HcapK) edge  (K)  ;
\draw (HK) edge  (H)  ;
\draw (HK) edge  (K)  ;
\draw (HK) edge  (G)  ;

ode (2) at (4,0) {$1$} ;

ode (H2) at (4,1.33) {$H = H\cap K$} ;

ode (K2) at (4,2.66) {$K = \langle H, K angle$} ;

ode (G2) at (4,4) {$G$} ;
\draw (2) edge  (H2)  ;
\draw (H2) edge  (K2)  ;
\draw (K2) edge  (G2)  ;

ode (3) at (8,0) {$1$} ;

ode (H2) at (8,1.33) {$K$} ;

ode (K2) at (8,2.66) {$H$} ;

ode (G2) at (8,4) {$G$} ;
\draw (3) edge  (H2)  ;
\draw (H2) edge  (K2)  ;
\draw (K2) edge  (G2)  ;

\end{tikzpicture}
\end{center}
(If $H 
ot \leq K$ and $ K 
ot \leq H$, or $H \leq K$, or $K \leq H$ respectively)

and some variants if $H \cap K = \{1\}$, $\langle H, K angle = G$, or both:
\bigskip
\begin{center}
\begin{tikzpicture}

ode (HcapK) at (0,1) {$1$} ;

ode (H) at (-1,2) {$H$} ;

ode (K) at (1,2) {$K$} ;

ode (HK) at (0,3) {$\langle H, K angle$} ;

ode (G) at (0,4) {$G$} ;
\draw (HcapK) edge  (H)  ;
\draw (HcapK) edge  (K)  ;
\draw (HK) edge  (H)  ;
\draw (HK) edge  (K)  ;
\draw (HK) edge  (G)  ;

ode (2) at (4,1) {$1$} ;

ode (HcapK2) at (4,2) {$H \cap K$} ;

ode (H2) at (3,3) {$H$} ;

ode (K2) at (5,3) {$K$} ;

ode (G2) at (4,4) {$G$} ;
\draw (2) edge  (HcapK2)  ;
\draw (HcapK2) edge  (H2)  ;
\draw (HcapK2) edge  (K2)  ;
\draw (H2) edge  (G2)  ;
\draw (K2) edge  (G2)  ;

ode (HcapK2) at (8,2) {$1$} ;

ode (H2) at (7,3) {$H$} ;

ode (K2) at (9,3) {$K$} ;

ode (G2) at (8,4) {$G$} ;
\draw (HcapK2) edge  (H2)  ;
\draw (HcapK2) edge  (K2)  ;
\draw (H2) edge  (G2)  ;
\draw (K2) edge  (G2)  ;

\end{tikzpicture}
\end{center}
\end{proof}

Exercise 2.5.1 (Sublattices with $H$ and $K$)

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Exercise 2.5.1 (Sublattices with $H$ and $K$)

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