All prime-ordered groups are cyclic

If a group $𝐺$ has prime order $𝑝 = | 𝐺 |$ , then it must be cyclic, #m/thm/group i.e. isomorphic to $ℤ + 𝑝$ .

Proof

Let $𝐺$ be a group of prime order $𝑝$ . By Lagrange's theorem, $𝐺$ only has subgroups of order $1$ and $𝑝$ . Since $𝑝 > 1$ , there exists $𝑎 \in 𝐺$ such that $𝑎 \neq 𝑒$ . Then $| 𝑎 | = 𝑝$ and therefore $⟨ 𝑎 ⟩ = 𝐺$ .

Clearly $ℤ + 𝑝$ is a simple group.

All prime-ordered groups are cyclic

See also