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 $|G|=p$. 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 $g\neq e$. Then $\langle g\rangle$ is a cyclic subgroup of $G$. $\langle g\rangle$ cannot be the trivial group since it contains $g$. So $|\langle g\rangle |=p$. So $G=\langle g\rangle$. 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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