I am unclear on how to solve problems like Dummit and Foote 2.5.5:
Use the given lattice to find all elements $x \in D_{16}$ such that $D_{16} = \langle x,s\rangle$ (there are 16 such elements $x$)
I can see solutions with $x=r,sr^3,sr^7,sr^5,sr$ using the process D&F describe for finding join of subgroups. However, I am lost on how to proceed with finding the other 11.
Since $D_{16}=\{1,r,r^2,...,r^7,s,sr,sr^2,...,sr^7\}$ has order 16, we at most must examine those 16 possibilities. By examining the joins in the lattice, we find the solutions $x=r,sr,sr^3,sr^5,sr^7$ with a join equal to the overall group. We can eliminate $r^2,r^4,sr^2,sr^4,sr^7$ since their joins are seen from the lattice to be proper subgroups. We can trivially eliminate 1 and s since the order of s is 2 so there is no way to form r from it. This leaves $r^3,r^5,r^6,r^7$. $(r^3)^3=(r^5)^5=(r^7)^7 = r$ so $<r^3>=<r^5>=<r^7>=<r>$. Also, $(r^6)^3 = r^2$ , $(r^2)^3 = r^6$ so $<r^6>=<r^2>$. Thus, $x=r^3,r^5,r^7$ are solutions while $x^6$ is not. Dummit and Foote have typo in my edition anyways. There are 8 solutions.