Gtagadarpcfullgamesetupexefreedownloadzip ((FULL)) – Kindergarten Wuppertal Zwergenburg

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Gtagadarpcfullgamesetupexefreedownloadzip ((FULL))

Gtagadarpcfullgamesetupexefreedownloadzip

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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 $ainmathbb{Z}_p$. Define
$$S_a={ninmathbb{Z}_pmid n-a
otinmathbb{Z}_p}.$$
Let $Ssubseteqmathbb{Z}_p$ 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 $ainmathbb{Z}_p$?

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?

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
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