Group of prime order is cyclic

Dependencies:

1. Group
2. Cyclic Group
3. Order of element divides order of group

Let $G$ be a group where $|G|$ is prime.

Since $|G| > 1$, $G$ has an element $g$ which is not identity. $\operatorname{order}(g) > 1$, because $g$ is not identity.

$\operatorname{order}(g)$ divides $|G|$ and $|G|$ is prime. Therefore, $\operatorname{order}(g) = |G|$. This means $g$ is a generator of $G$.

Therefore, a group of prime order is cyclic and all non-identity elements are generators.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 15

Transitive dependencies:

1. Group
2. Coset
3. Size of coset equals size of subset
4. Identity of a group is unique
5. Order of element in finite group is finite
6. Subgroup
7. Inverse of a group element is unique
8. gH = H iff g in H
9. Two cosets are either identical or disjoint
10. Lagrange's Theorem
11. Conditions for a subset to be a subgroup
12. Cyclic Group
13. Integer Division Theorem
14. Order of cyclic subgroup is order of generator
15. Order of element divides order of group