Characterization of cyclic Schur groups

Abstract:

A finite group $G$ is called a Schur group if any Schur ring over $G$ is associated in a natural way with a subgroup of $\mathrm {Sym}(G)$ that contains all right translations. It was proved by R. Pöschel (1974) that, given a prime $p\ge 5$, a $p$-group is Schur if and only if it is cyclic. We prove that a cyclic group of order $n$ is Schur if and only if $n$ belongs to one of the following five families of integers: $p^k$, $pq^k$, $2pq^k$, $pqr$, $2pqr$ where $p$, $q$, $r$ are distinct primes, and $k\ge 0$ is an integer.