Problem: Find all subgroups of , draw the subgroup diagram.

Corollary:

If is a generator of a finite cyclic group of order , then the other generators G are the elements of the form , where r is relatively prime to n.

I'm following this problem in the book. It says that by corollary 6.16 (what I wrote above), the elements 1, 5, 7, 11, and 17 are all generators of .

These elements (1, 5, 7 ...) are from using the generator <1>, correct? And therefore (1, 5, 7...) will not only be subgroups of but will actually equal , correct?

Then the example starts with <2>, which is the next generator after <1>, which forms the subgroup {0, 2, 4, 6, 8, 10, 12, 14, 16}. It then goes on to say that <2> "generates elements of the form , where is relatively prime to 9, namely, h=1,2,4,7,8 so h2=2,4,8,10,14,16. What exact is the purpose of that bolded statement? Is it just used to save time, since we can now say by that corollary, is a generator because 4 is relatively prime to n=9?

It continues, saying that the element 6 of <2> generates {0, 6, 12}, and 12 is also a generator of this group. Why exactly are we looking at <2>? Are we methodically going through each generator and looking at all of it's elements that we haven't used as generators yet, to use as generators?

Continuing, it says we have found all the subgroups generated by 0,1,2,4,5,6,7,8,10,11,12,13,14,16,17. This just leaves 3, 9 and 15 to consider.

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Now I'm assuming since we've already seen 0, 6 and 12, we are only concerned with 3, 9, and 15. It says that 15 can also generate this group, which is easy to see. So we ignore it in just the diagram? Or just ignore it altogether?

9 is the last generator left unexplored, so <9>={0,9}.

The subgroup then looks like this:

<1> to <2>, <3>

<2> to <1>, <6>

<3> to <6>, <9>, <1>

<6> to <2>, <3>, <0>

<9> to <3>, <6>, <0>

<0> to <6>, <9>

What exactly are these nodes? Are they the generators that we didn't ignore in the process of finding all of the subgroups? Why aren't we using any of the generators (besides 1) found in the first part of the problem? What exactly was the point of finding those generators in the first part of the problem?

Any help is appreciated.