Equivalence of definitions of group of prime order

A group of prime order, or cyclic group of prime order, is any of the following equivalent things:

1. It is a cyclic group whose order is a prime number.
2. It is isomorphic to the quotient of the group of integers by a subgroup generated by a prime number.
3. It is the additive group of a finite prime field (note that we have to say finite because "prime field" also includes the field of rational numbers, the prime field of characteristic zero).

If $p$ denotes the order of the group, then the cyclic group of order $p$ is denoted $C_p$, $\mathbb{Z}/p\mathbb{Z}$, ${\mathbb{Z}}_p$ or ${\mathbb{F}}_p$.

This page is dedicated to showing that these definitions are equivalent.

If $p$ is prime, then a group of order $p$ is cyclic

Suppose $p$ is prime and $G$ is a group such that $\left|G\right|=p$. Then by Lagrange's theorem and the fact $p$ is prime, any subgroup of $G$ must have order $1$ or $p$. Take $g\in G$ such that $g\ne e$. Then $⟨g⟩$ is a cyclic subgroup of $G$. $⟨g⟩$ cannot be the trivial group since it contains $g$. So $|⟨g⟩|=p$. So $G=⟨g⟩$. So $G$ is cyclic.

The quotient of the group of integers by a subgroup generated by a prime number is cyclic

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It is the additive group of a finite prime field

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