On Schurity of Finite Abelian 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 Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ℤ₃ × ℤ_{3 ^k} or ℤ₃ × ℤ₃ × ℤ _p where k ≥ 1 and p is a prime. In addition, we prove that ℤ₂ × ℤ₂ × ℤ _p is a Schur group for every prime p.

Key Words:

• Permutation group
• Schur group
• Schur ring

2010 Mathematics Subject Classification:

• 20B25
• 05E30

Notes

The terms “schurian S-ring” and “Schur group” do not seem quite satisfactory. However, they were used not only in papers [1, 3, 5, 6] already published, but also in several forthcoming article, see remark in the end of Introduction.

Over a group of order at most 41.

This can be done via an appropriate decomposition E ₄ = ℤ₂ × ℤ₂.

In fact, the case when P is not an 𝒜-group is dual to Case 1 in the sense of the duality theory of S-rings over an abelian group, see [4]; here, we prefer a direct proof.

Any S-ring over H is of the form Cyc(K, H) where K = Aut(H) ≅ Sym(3) or K is a subgroup of order 2 in Aut(H) or K = 1.

Communicated by A. Turull.