Use the subgroup lattice of Dih(16) to answer questions about subgroups

In each of 1-4 list all the subgroups of D_{16} that satisfy the given condition.

  1. Subgroups that are contained in \langle sr^2, r^4 \rangle
  2. Subgroups that are contained in \langle sr^7, r^4 \rangle
  3. Subgroups that contain \langle r^4 \rangle
  4. Subgroups that contain \langle s \rangle

We assume the subgroup lattice of D_{16}.

  1. \langle sr^2,r^4 \rangle contains \langle sr^2,r^4 \rangle, \langle sr^6 \rangle, \langle sr^2 \rangle, \langle r^4 \rangle, and 1.
  2. Note that \langle sr^7,r^4 \rangle = \langle sr^3,r^4 \rangle since sr^7 = sr^3 \cdot r^4 and sr^3 = r^4 \cdot sr^7. Thus the subgroups contained in \langle sr^7,r^4 \rangle are \langle sr^3,r^4 \rangle, \langle r^4 \rangle, \langle sr^3 \rangle, \langle sr^7 \rangle, and 1.
  3. \langle r^4 \rangle is contained in \langle r^4 \rangle, \langle sr^2,r^4 \rangle, \langle s,r^4 \rangle, \langle r^2 \rangle, \langle sr^3, r^4 \rangle, \langle sr^5,r^4 \rangle, \langle s,r^2 \rangle, \langle r \rangle, \langle sr,r^2 \rangle, and D_{16}.
  4. \langle s \rangle is contained in \langle s \rangle, \langle s,r^4 \rangle, \langle s,r^2 \rangle, and D_{16}.
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  • Samantha  On September 30, 2010 at 5:13 pm

    I believe that for number 1 < r^4 > is also contained within < sr^2,r^4 >

    • nbloomf  On September 30, 2010 at 6:58 pm

      You’re right. Thanks!

  • Jake  On December 7, 2011 at 7:05 am

    I think \langle r^{4} \rangle is contained in \langle s, r^{4} \rangle and not \langle sr^{4} \rangle

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