세미나/포럼
[세미나/포럼] [2019.03.15.금] 수학과 위상조합세미나 개최 안내
- 세미나/포럼
- 교내
- 박수현
- 2019-03-11
안녕하세요. 아주대 위상조합 세미나를 아래와 같이 개최하고자 합니다.
많은 참여부탁드립니다.
제목 : A characterization of the groups whose proper Schur rings are commutative
연사 : 조은경 박사 (한국외국어대학교)
일시 : 3월 15일 금요일 오후 1:30 ~ 2:30
장소 : 팔달관 621호
초록 :
Let $\mathds{Z}[G]$ be the group ring of a group $G$ over $\mathds{Z}$. A Schur ring $\mathscr{A}$ over $G$ is a subring of $\mathds{Z}[G]$ which is determined by a certain partition of $G$ (called a Schur partition). An $\mathscr{A}$-subgroup is a subgroup of $G$ that is a union of some elements from the Schur partition corresponding to $\mathscr{A}$. We define a Schur ring $\mathscr{A}$ to be {\it Dedekind} if the formal sum of every $\mathscr{A}$-subgroup is contained in the center of $\mathscr{A}$. 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 {\it proper} if it is not the group ring. We prove in this talk 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.