# Prime order groups are cyclic

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 $$p$$.

# Proof

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.

Parents:

• Group

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