1. Characterization of cyclic Schur groups
- Author
-
Ilya Ponomarenko, Sergei Evdokimov, and István Kovács
- Subjects
Discrete mathematics ,Mathematics::Combinatorics ,Algebra and Number Theory ,Applied Mathematics ,Schur's lemma ,Perfect group ,Primitive permutation group ,Schur algebra ,Schur's theorem ,Covering groups of the alternating and symmetric groups ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,05E30, 20B25 ,Mathematics::Representation Theory ,Analysis ,Schur multiplier ,Schur product theorem ,Mathematics - Abstract
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. P\"oschel (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 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 $k\ge 0$ is an integer., Comment: the second version; the proof was substantially improved; 29 pages
- Published
- 2014