 von devtal
VerĂ¶ffentlich: 10. September 2022 (vor 7 Monaten )
Kategorie

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 \$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?

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