0
$\begingroup$

Given the subgroup lattice of the dihedral group $D_{8}$, find find all pairs of elements that generate $D_{8}$. There should be 8 pairs.

The given lattice is the image below:

Subgroup lattice of dihedral group of order 8

I came to following set of pairs:

$\left \langle r, s \right \rangle$, $\left \langle r^2, s \right \rangle$, $\left \langle rs, r \right \rangle$, $\left \langle r^3s, r \right \rangle$, $\left \langle s, rs \right \rangle$, $\left \langle s, r^3s \right \rangle$, $\left \langle r^2s, rs \right \rangle$, $\left \langle r^2s, r^3s \right \rangle$

and

$\left \langle s, r^3 \right \rangle$, $\left \langle rs, r^3 \right \rangle$, $\left \langle r^3s, r^3 \right \rangle$, $\left \langle r^3, r^2s \right \rangle$

But I am not sure if this is right.

Also, how do you analyse the lattice step by step to reach the solution? Finding out first 8 was simple, because I had to find single-element subgroups and combine them to form pairs. However, the rest 4, I derived from found ones by searching for equivalent subgroups.

This is exercise 2.5.4 in Dummit & Foote "Abstract Algebra" (3rd ed.).

$\endgroup$
3
$\begingroup$

It's $12$, really.

You have to choose first a pair of distinct maximal subgroups. And then one element from each of these maximal subgroup, neither of which elements lies in their intersection.

For instance, if you take the pair of maximal subgroups $$ \langle s, r^2 \rangle, \quad \langle r \rangle $$ you can choose either $s$ or $s r^2$ from the first subgroup, and either $r$ or $r^3$ from the second one, This makes for $4$ generating pairs.

As there are three pairs of maximal subgroups, and for distinct maximal subgroups $M, N$ you have $\lvert M \setminus M \cap N \rvert = 4 - 2 = 2$, you get $2 \cdot 2 \cdot 3 = 12$ (unordered) pairs.

$\endgroup$
  • $\begingroup$ You are right. 12 pairs in total. I mistyped... $\endgroup$ – taro Jan 12 '16 at 8:34
  • $\begingroup$ Why $r$ or $r^3$ from $<r>$? How can we tell from the lattice and not properties of $D_8$ directly that $r \in <r_3,s>$ so that it truly generated $D_8$? $\endgroup$ – topoquestion Sep 7 '16 at 14:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.