Isomorphism between Group of Units of Ring of Integers Modulo p^n and C((p-1)p^(n-1))

Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $p$ be an odd prime.

Let $R = \struct {\Z / p^n \Z, +, \times}$ be the ring of integers modulo $p^n$.

Let $U = \struct {\paren {\Z / p^n \Z}^\times, \times}$ denote the group of units of $R$.

Let $C_{(p - 1)p^{n - 1}}$ be be cyclic group of order $(p - 1)p^{n - 1}$.

Then $U$ is isomorphic to $C_{(p - 1)p^{n - 1}}$.

Proof

The case $n = 1$ is proved in Ring of Integers Modulo Prime is Field.

Suppose $n \ge 2$.

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Source

• 1801: Carl Friedrich Gauss: Disquisitiones Arithmeticae, arts. 84–89