How to find all the subgroups of $\mathbb{Z}_4$

Question

How to find all the subgroups of $\mathbb{Z}_4$

I know that $\mathbb{Z}_4$ = {$\bar{0},\bar{1},\bar{2},\bar{3}$}. Also how about working out the subgroups for a much larger number e.g. $\mathbb{Z}_{20}$

Answer 1

Actually, there is no exact formula. You can find these subgroups by trial and error and some known theorems which help to find orders, like Lagrange's theorem, Cauchy's theorem and Sylow theorems. Try to combine elements and see what you get.

Answer 2

Hint : $(i)$ $\Bbb Z _m$ is a cyclic group of order m .

$(ii)$ There is a well-known result : If $G$ is a cyclic group of finite order (say, $n$), then for each positive integer $d$ so that $d|n$, $\exists$a unique subgroup of $G$ of order $d$. (In fact, you can try to prove this delicate, yet not-so-difficult result).

$(iii)$ Combining $(i)$ and $(ii)$ , find all possible orders of subgroups of $G$.

$(iv)$ Any subgroup of a Cyclic group is Cyclic.

$(v)$ In any finite cyclic group there exists an element with order equal to the order of the group.

Best Of Luck!