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 $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this
paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to $Z_3times Z_{3^k}$ or $Z_3times Z_3times Z_p$ where $kge 1$ and $p$ is a prime. In addition, we prove that $Z_2times Z_2times Z_p$ is a Schur group for every prime $p$.
A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a permutation group on the set $G$ containing the regular subgroup of all right translations. It was proved by R. Poschel (1974) that given a prime $
pge 5$ a $p$-group is Schur if and only if it is cyclic. We prove that a cyclic group of order $n$ is a Schur group if and only if $n$ belongs to one of the following five (partially overlapped) families of integers: $p^k$, $pq^k$, $2pq^k$, $pqr$, $2pqr$ where $p,q,r$ are distinct primes, and $kge 0$ is an integer.
