Exercise 2.5.5 (Find all elements $x \in D_{16}$ such that $D_{16} = \langle x ,s \rangle$)

Question

Use the given lattice to find all elements $x \in D_{16}$ such that $D_{16} = \langle x ,s \rangle$ (there are $8$ (*) such elements $x$)

\bigskip
(*) Obvious misprint in D.F., which write ``$16$ such elements $x$'' even though $D_{16}$ has $16$ elements.

Richard Ganaye · Accepted Answer

\begin{proof} 
We must be cautious, because the lattice deals with groups, and not elements.

First we read on this lattice that the upper limit of $\langle s angle$ and $\langle r angle$ is $D_8$. Moreover $\langle r angle = \langle r^3 angle = \langle r^5 angle =\langle r^7 angle $, because $n \wedge 8 = 1 \iff n \in \{1,3,5,7\}$. Hence
$$D_8 = \langle r, sangle = \langle r^3, s angle = \langle r^5, s angle =\langle r^7, s angle.$$
(But if $k$ is even $\langle r^k,s angle \leq \langle s, r^2angle 
e D_{16}$.)

Now we test the elements $sr^k,\ 0 \leq k < 8$. We read on the diagram that the upper limit of $\langle s angle$ with $\langle sr^3 angle$ (or with $\langle sr^7 angle$, $\langle sr^5 angle$,$\langle sr angle$) is $D_8$, so
$$D_8 = \langle  sr^3, s angle, \langle  sr^7, sangle, \langle sr^5, s angle, \langle  sr, s angle.$$
(But if $k$ is even $\langle sr^k, s angle \leq \langle s,r^2 angle$.)

This gives exactly $8$ elements $x$ such that $D_{16} = \langle x ,s angle$:
\begin{align*}
D_{16} &= \langle r, sangle = \langle r^3, s angle = \langle r^5, s angle =\langle r^7, s angle\
&= \langle  sr^3, s angle, \langle  sr^7, sangle, \langle sr^5, s angle, \langle  sr, s angle.
\end{align*}
\end{proof}
With Sagemath:
\begin{verbatim}
sage: D16 = DihedralGroup(8); D16.order()
16
sage: [r,s] = D16.gens(); r,s
((1,2,3,4,5,6,7,8), (1,8)(2,7)(3,6)(4,5))
sage: def my_word_problem(x, n):
....:      """ Only for Dihedral Group D_2n"""
....:      u, v = -1,-1
....:      for i in range(n):
....:          for j in range(2):
....:              if  r^i * s^j  == x:
....:                  u, v = i, j
....:      if u == 0:
....:          f = ""
....:      elif u == 1:
....:          f = "r"
....:      else:
....:          f = "r^{}".format(u)
....:      if v == 0:
....:          g = ""
....:      elif v == 1:
....:          g = "s"
....:      else:
....:          g = "s^{}".format(v)
....:      if u==0 and v == 0:
....:          f = '1'
....:      return( g + f)
....:  
sage: my_word_problem(r*s, 8) #r * s is sr in Dummit, Foote
'sr'
sage: liste = []
....: for x in D16:
....:      if D16.subgroup([x,s]).order() == 16:
....:          liste.append(x)
....: liste
....:

[(1,2,3,4,5,6,7,8),
 (1,7)(2,6)(3,5),
 (2,8)(3,7)(4,6),
 (1,4,7,2,5,8,3,6),
 (1,8,7,6,5,4,3,2),
 (1,5)(2,4)(6,8),
 (1,3)(4,8)(5,7),
 (1,6,3,8,5,2,7,4)]
sage: for x in liste:
....:     print(my_word_problem(x, 8))
....:     
r
sr
sr^7
r^3
r^7
sr^3
sr^5
r^5
\end{verbatim}

Exercise 2.5.5 (Find all elements $x \in D_{16}$ such that $D_{16} = \langle x ,s \rangle$)

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Exercise 2.5.5 (Find all elements $x \in D_{16}$ such that $D_{16} = \langle x ,s \rangle$)

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