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Q: Is there a way to decide if elements of subsets of $\mathbb{Z}_p$ are invertible? Let $p$ be a prime number, let $\mathbb{Z}_p$ be the ring of the $p$-adic integers and let $a\in\mathbb{Z}_p$. Define $$S_a=\{n\in\mathbb{Z}_p\mid n-a ot\in\mathbb{Z}_p\}.$$ Let $S\subseteq\mathbb{Z}_p$ be any set with $p^n$ elements. Question: Is there a simple way to decide if all elements of $S_a$ are invertible in $\mathbb{Z}_p$, for any $a\in\mathbb{Z}_p$? I wonder if this is somehow related to the following question: Question: Is there a prime $p$ such that $\mathbb{Z}_p$ is not an injective $\mathbb{Z}$-module? A: These are all invertible since otherwise you will get some $x \in \mathbb{Z}/p$ such that $x^2=0$ and $x(n-a) =0$. More generally, let $S$ be any subset of $\mathbb{Z}_p$ of size $p^n$. Assume $S$ is not invertible. Then $S \subseteq \mathbb{Z}_p \setminus \{0\}$ and thus $\mathbb{Z}_p \setminus S \subseteq \mathbb{Z}_p \setminus \{0\}$. This implies that $\mathbb{Z}_p \setminus S$ is a subgroup of $\mathbb{Z}_p$ of size $p^{n-1}$ and thus the nilpotency degree of $\mathbb{Z}_p / \mathbb{Z}_p \setminus S$ is at least $n-1$ (a nilpotent group of order $p^n$ has nilpotency degree $\leq n-1$). By the Higman-N 6d1f23a050
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