Exercise 2.5.19 (Find $N_{D_{16}}(\langle s, r^4 \rangle)$)

Question

Use the lattice to help find $N_{D_{16}}(\langle s, r^4 \rangle)$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Since $Z(D_{16}) = \langle r^4 angle$ (Exercises 1.2.4 and 2.5.7), then $s$ commutes with $r^4$, and with $s$. Therefore $s \in N_{D_{16}}(\langle s, r^4 angle)$.
Moreover, 
$$r^2 s r^{-2} = r^4 s \in \langle s, r^4 angle,$$
and of course $r^2 r^4 r^{-2} = r^4 \in \langle s, r^4 angle$. Therefore $r^2 \in N_{D_{16}}(\langle s, r^4 angle)$, so
$$\langle s, r^2 angle \leq N_{D_{16}}(\langle s, r^4 angle).$$
The lattice of subgroups of $D_{16}$ (see Example 4) shows that $N_{D_{16}}(\langle s, r^4 angle) = \langle s, r^2 angle$ or $N_{D_{16}}(\langle s, r^4 angle) = D_{16}$.

In the last case, $\langle s, r^4 angle$ would be a normal subgroup, but
$$rs r^{-1} = sr^{-2} = s r^6 
ot \in \langle s, r^4 angle = \{1, s, r^4, sr^4\}.$$
Hence
$$N_{D_{16}}(\langle s, r^4 angle) = \langle s, r^2 angle.$$
\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 wp(x):
....:      """ Only for Dihedral Group D_16"""
....:      n = 8
....:      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: wp(r*s) # r * s is sr in Dummit, Foote
'sr'
sage: H = D16.subgroup([s, r^4])
sage: K = D16.normalizer(H)
....: rep = ''
....: for x in K.list():
....:     rep = rep + wp(x) + ' ' 
....: print(rep)
....:  
1 sr^6 r^4 s sr^2 r^6 r^2 sr^4 
sage: D16.normalizer(H) == D16.subgroup([s,r^2])
True
\end{verbatim}

Exercise 2.5.19 (Find $N_{D_{16}}(\langle s, r^4 \rangle)$)

Answers

Comments

Exercise 2.5.19 (Find $N_{D_{16}}(\langle s, r^4 \rangle)$)

Answers

Comments

Add answer