Determining subgroups of a finite field and their elements

Question

I'm studying cryptography and while reading some lecture notes I found the following:

$F$*₃₇ has subgroups of order 2 ({2⁰ , 2¹⁸}), 3 ({2⁰ , 2¹² , 2²⁴}), 4, 6, 9, 12, and 18.

1. How to determine that the subgroups are of order 2, 3, 4, 6, 9, 12, and 18?
2. How to determine the elements of those subgroups? (for example ({2⁰ , 2¹⁸}) in the first one.) I know in this one that the base is 2 because the field generator is 2, but where did those powers come from?

Thank you.

Answer 1

You are trying to use too much information!

You are trying to study the cyclic group of 36 elements and you have chosen a generator of the group. Once you know this, the fact it originated from a finite field doesn't matter at all -- you should ignore that piece of information entirely, and just focus on the fact you're studying a cyclic group.