Prime order groups are cyclic
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.
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.