Prime order groups are cyclic

by Patrick Stevens Jun 17 2016 updated Jun 20 2016

This is the first step on the road to classifying the finite groups.

Let $~$G$~$ be a Group whose order is equal to $~$p$~$, a Prime number. Then $~$G$~$ is isomorphic to the Cyclic group $~$C_p$~$ of order $~$p$~$.


Pick any non-identity element $~$g$~$ of the group.

By Lagrange's theorem, the subgroup generated by $~$g$~$ has size $~$1$~$ or $~$p$~$ (since $~$p$~$ was prime). But it can't be $~$1$~$ because the only subgroup of size $~$1$~$ is the trivial subgroup.

Hence the subgroup must be the entire group.