I'm failing to see the logic behind what makes up a subgroup. I understand the requirements of a subgroup (associativity, identity etc….) but I don't actually know how to find the subgroups.
I think it has something to do with "getting back to the identity", but I may be wrong? I know that the identity is a subgroup and the whole group is a subgroup. That's all.
Any help would be appreciated!
Best Answer
Edit: Written this assuming you've taken a course on group theory.
Finding all subgroups of large finite groups is in general a very difficult problem. Usually, I'd start with Lagrange's theorem to find possible orders of subgroups.
Next, you know that every subgroup has to contain the identity element. Then you can start to work out orders of elements contained in possible subgroups - again noting that orders of elements need to divide the order of the group.
Since you've added the tag for cyclic groups I'll give an example that contains cyclic groups.
Consider the dihedral group $D_n = \langle r,s \mid r^n = s^2 = e, srs = r^{-1} \rangle$ where $e$ is the identity. It has order $2n$ and so the order of subgroups must divide $2n$. One such example is the subgroup $\langle r\rangle = e, r, r^2, r^3, ..., r^{n-1} $ which is clearly isomorphic to $\mathbb{Z}_n$.