An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true.

Is there a $\mathbb Z$-order $\Lambda$ in $\mathbb Q[G]$ which contains $\mathbb Z[G]$ and satisfies the following two conditions

1) $\Lambda\otimes_\mathbb Z \mathbb Z_p$ is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ if $p\in S$

2) $\Lambda\otimes_\mathbb Z \mathbb Z_p=\mathbb Z_p[G]$ if $p\notin S$?

Here $\mathbb Q_p$ (resp. $\mathbb Z_p$) is the $p$-adic completion of $\mathbb Q$ (resp. $\mathbb Z$).

A proof or reference will be gratefully accepted.

Many thanks.

Answer

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Source : Link , Question Author : eddie , Answer Author : Community

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