[대수학] 소수 오더를 갖는 그룹은 씨클릭(cyclic, 순환군) 이다.

" Every group of prime order is cyclic."

 Let G be of prime order p , and let a be an element of G other than the identity.

Consider the cyclic subgroup <a>.
Since a ≠ e , |a| = |<a>| ≥ 2 .

By Lagrange's theorem, |<a>| must divide |G| = p

There are only two numbers dividing a prime number, that is 1 and p itself.
Since |<a>| ≠ 1 , |<a>| = p .

Thus G = <a> , so G is cyclic.