A Characterization of the Groups Whose Proper Schur Rings are Commutative = 진 슈어 환이 모두 교환 가능한 군에 관하여

다국어 초록 (Multilingual Abstract)

Let Z[G] be the group ring of a group G over Z. A Schur ring over G is a subring
of Z[G] which is determined by a certain partition of G. The class of Boolean groups is
related to the class of symmetric Schur rings in the sense that all the Schur rings over
a Boolean group are symmetric. The class of abelian groups is related to the class of
commutative Schur rings in the sense that all the Schur rings over an abelian group are
commutative. We define a Schur ring A to be Dedekind if the formal sum of every A -
subgroup is contained in the center of A . Then the class of Dedekind groups is related
to the class of Dedekind Schur rings in the sense that all the Schur rings over a Dedekind
group are Dedekind Schur rings. A Schur ring is proper if it is not the group ring. We
prove in this thesis that all the proper Schur rings over a group G are Dedekind Schur
rings if and only if G is a Dedekind group or a dihedral group of order 8 or 2 times a
Fermat prime. As a corollary of this result, we prove that all the proper Schur rings over
a group G are commutative if and only if G is an abelian group, the quaternion group, or
a dihedral group of order 8 or 2 times a Fermat prime. Also, we prove that all the proper
Schur rings over a group G are symmetric if and only if G is a Boolean group or a cyclic
group of order 4 or a Fermat prime.

목차 (Table of Contents)

• Contents
• Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
• 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
• 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
• 2.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
• 2.2 Schur Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
• 3 Dedekind Schur Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 16
• 3.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . 16
• 3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
• 4 Non-Dedekind and Non-Dihedral Groups . . . . . . . . . . . . . . 30
• 5 Commutative Schur Rings and Symmetric Schur Rings . . . . . 39
• Abstract(Korean) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43