I know how to find all the subgroups of $n$ for a specific values of $n$, for example for $n=18$, the subgroups of $\Bbb{Z}_{18}$ are:

<1>={0,1, ... ,17}
<2>={0,2, ... ,16}
<3>={0,3, ... ,15}

But, how do we draw the subgroup diagram of this? What are the steps? Here is the subgroup diagram that is given:enter image description here Can anyone explain how to draw this? Thanks


notice it's exactly the same as the divisibility lattice (upside down):

9   6
  3   2

form a statement about the relation between the subgroup lattice of groups and the divisibility lattice of numbers and prove it.

  • $\begingroup$ You should probably mention that you are using the number $n$ to represent the subgroup $\mathbb{Z}_n$, since this isn't what a student should write down as the subgroup lattice in their homework! $\endgroup$ – Tara B Mar 16 '13 at 13:24
  • $\begingroup$ @TaraB, I'm using the number n to represent the number n I've tried to make that clearer thanks for the comment. $\endgroup$ – user58512 Mar 16 '13 at 13:44
  • 1
    $\begingroup$ Oh! I see. Your picture is of the divisibility lattice rather than the subgroup diagram! I did wonder about the 'upside down'. $\endgroup$ – Tara B Mar 16 '13 at 13:48

To draw the subgroup diagram, you need to know not only what the subgroups are, but also the containment relationship between them. This is easy to read off from your list of subgroups, and then you get the picture in caveman's answer.

In general you might have quite a large cyclic group, and so you don't necessarily want to write down all the elements of all the subgroups. But if your cyclic group is $\langle a\rangle$ and $a$ has order $m$, then you know that each subgroup is generated by some $\langle a^k\rangle$, where $k$ is a divisor of $m$. When will $\langle a^k\rangle$ be a subgroup of $\langle a^l\rangle$?


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