$U/U_3 \equiv (\mathbb{Z}/27\mathbb{Z})^{\times}$

Question

I was reading this: What is the group structure of 3-adic group of the cubes of units?, but I do not understand why $U/U_3 \equiv (\mathbb{Z}/27\mathbb{Z})^{\times}$

Answer 1

Let $U=\Bbb Z_p^\times$, the group of units in the $p$-adic integers. Let $U_k=\{u\in U: u\equiv1\pmod{p^k}\}$ for $k\in\Bbb N$. Then $U/U_k\cong(\Bbb Z/p^k\Bbb Z)^\times$. The reason is that mapping $u\in U$ to $u$ modulo $p^k$ is a surjective group homomorphism to $(\Bbb Z/p^k\Bbb Z)^\times$ with kernel $U_k$.