- #1

- 1,462

- 44

## Homework Statement

Show that for every subgroup ##H## of cyclic group ##G##, ##H = \langle g^{\frac{|G|}{|H|}}\rangle##.

## Homework Equations

## The Attempt at a Solution

At the moment the most I can see is that ##|H| = |\langle g^{\frac{|G|}{|H|}}\rangle|##. This is because if ##(g^{\frac{|G|}{|H|}})^p = e## for some value of ##p < |H|##, we would have that ##g^q = e## for some ##q < |G|##, which is a contradiction.

But I'm not seeing why they must be the same subgroup. I was going to try to prove that if two subgroups of cyclic group have the same order, then they must be the same group. If I did this I think I would first need to prove that every subgroup of a cyclic group is cyclic.

Last edited: