Andrew Misseldine
Any Schur ring is uniquely determined by a partition of the elements of the group. In this paper we present a general technique for enumerating Schur rings over cyclic groups using traditional Schur rings. We also survey recent efforts to enumerate Schur rings over cyclic groups of specific orders.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The generalized wreath product of permutation groups is introduced. By means of it we study the schurity problem for S-rings over a cyclic group $G$ and the automorphism groups of them. Criteria for the schurity and non-schurity of the generalized wreath product of two such S-rings are obtained. As a byproduct of the developed theory we prove that $G$ is a Schur group whenever the total number $Ω(n)$ of prime factors of the integer $n=|G|$ is at most 3. Moreover, we describe the structure of a non-schurian S-ring over $G$ when $Ω(n)=4$. The latter result implies in particular that if $n=p^3q$ …
The famous Burnside-Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latters we introduce Schur rings over a finite commutative ring and prove an analog of this statement for them. Besides, the finite local commutative rings are characterized in the permutation group terms.
Counting Schur Rings over Cyclic Groups of Semi-prime Order
本文继续研究循环群上的Schur环计数问题，重点分析半素数阶循环群pq及4p阶循环群的结构与性质。
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 $Z_3\times Z_{3^k}$ or $Z_3\times Z_3\times Z_p$ where $k\ge 1$ and $p$ is a prime. In addition, we prove that $Z_2\times Z_2\times Z_p$ is a Schur group for every p…
Any Schur ring is uniquely determined by a partition of the elements of the group. An open question in the study of Schur rings is determining which partitions of the group induce a Schur ring. Although a structure theorem is available for Schur rings over cyclic groups, it is still a difficult problem to count all the partitions. For example, Kovacs, Liskovets, and Poschel determine formulas to count the number of wreath-indecomposable Schur rings. In this paper we solve the problem of counting the number of all Schur rings over cyclic groups of prime power order and draw some parallels with …
Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed, using the canonical character of $B$. We also investigate the Frobenius-Schur indicators of the non-exceptional characters in $B$. For a finite group which has cyclic Sylow $p$-subgroups, we show that the number of irreducible characters with Frobenius-Schur indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements i…
Enumeration of skew morphisms of cyclic $2$-groups
本文研究了有限循环2-群的斜映射的计数问题，推导出递推公式并完成对所有循环p-群斜映射的完整分类与计数。
For the direct product $\cZ\times \cZ_3$ of infinite cyclic group $\cZ$ and a cyclic group $\cZ_3$ of order $3$, the schur rings over it are classified. In particular, all the schur rings are proved to be traditional.
Schur rings are a type of subring of the group ring that is spanned by a partition of the group that meets certain conditions. Past literature has exclusively focused on the finite group case. This paper extends many classic results about Schur rings to the infinite groups, including Leung-Man's classification of Schur rings over finite cyclic groups. Additionally, this classification will be extended to all torsion-free locally cyclic groups.
In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central idempotents of lattice Schur rings are completely determined. For a general Schur ring $S$, $S$ contains a maximal lattice Schur ring, whose central, primitive idempotents form a system of pairwise orthogonal, central idempotents in $S$. We show that if $S$ is a Schur ring with rational coefficients over a cyclic group, then these idempotents are always primitiv…
In 1993, Muzychuk \cite{muzychuk} showed that the rational Schur rings over a cyclic group $Z_n$ are in one-to-one correspondence with sublattices of the divisor lattice of $n$, or equivalently, with sublattices of the lattice of subgroups of $Z_n$. This can easily be extended to show that for any finite group $G$, sublattices of the lattice of characteristic subgroups of $G$ give rise to rational Schur rings over $G$ in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian $p$-group, for a…
An explicit formula for the Schur multiplier of the group of unitriangular matrices over products of cyclic rings $\ZZ/m\ZZ$ and $\ZZ$ is derived. We use it to provide presentations of the corresponding covering groups and touch upon the question of finding their nonisomorphic representatives.
Classification of abelian Schur groups I
本文研究了abelian Schur groups的分类，证明了某些直接积群是Schur群，同时证明了其他群非Schur。
In this paper the relationship between $S$-subgroups in $\Z_{2}^{n}$ and binary codes is shown. If the codes used are both $P(T)$-codes and $G$-codes, then the $S$-subgroup is free. The codes constructed are cyclic, decimated or symmetric and the $S$-subgroups obtained are free under the action the cyclic permutation subgroup, invariants under the action the decimated permutation subgroup and symmetric under the action of symmetric permutation subgroup, respectively. Also it is shows that there is no codes generating whole $\Z_{2}^{n}$ in any $\G_{n}(a)$-complete $S$-set of the $S$-ring $\math…
On schurity of dihedral groups
本文研究二面体群的 Schur 性，证明广义二面体 Schur 群必为二面体群，并对含特定素数条件的阶 $2p$ 二面体群给出 Schur 性证明及 $S$-环分类。
We study the notion of formal-duality over finite cyclic groups of prime power order as introduced by Cohn, Kumar, Reiher and Schürmann. We will prove that for any cyclic group of odd prime power order, as well as for any cyclic group of order $2^{2l+1}$, there is no primitive pair of formally-dual subsets. This partially proves a conjecture, made by the priorly mentioned authors, that the only cyclic groups with a pair of primitive formally-dual subsets are $\{0\}$ and $\mathbb{Z}/4\mathbb{Z}$.
On Schur Rings over Infinite Groups III
本文进一步研究无限群上的Schur环性质，特别关注虚循环群Z×Z_p，证明了Fermat和安全素数下的完整分类定理，指出所有Schur环均为传统类型，并探讨了Schur环与Z_p上差集划分的类比。
An $S$-ring (Schur ring) is called separable with respect to a class of $S$-rings $\mathcal{K}$ if it is determined up to isomorphism in $\mathcal{K}$ only by the tensor of its structure constants. An abelian group is said to be separable if every $S$-ring over this group is separable with respect to the class of $S$-rings over abelian groups. Let $C_n$ be a cyclic group of order $n$ and $G$ be a noncylic abelian $p$-group. From the previously obtained results it follows that if $G$ is separable then $G$ is isomorphic to $C_p\times C_{p^k}$ or $C_p\times C_p\times C_{p^k}$, where $p\in \{2,3\}…
A Schur ring (S-ring) over a group $G$ is called separable if every of its similaritities is induced by isomorphism. We establish a criterion for an S-ring to be separable in the case when the group $G$ is cyclic. Using this criterion, we prove that any S-ring over a cyclic $p$-group is separable and that the class of separable circulant S-rings is closed with respect to duality.
Two basic results on the S-rings over an abelian group are the Schur theorem on multipliers and the Wielandt theorem on primitive S-rings over groups with a cyclic Sylow subgroup. None of these theorems is directly generalized to the non-abelian case. Nevertheless, we prove that they are true for the central S-rings, i.e., for those which are contained in the center of the group ring of the underlying group (such S-rings naturally arise in the supercharacter theory). We also generalize the concept of a B-group introduced by Wielandt, and show that any Camina group is a generalized B-group wher…
Enumeration of virtual quandles up to isomorphism
本文通过计算机搜索对虚拟quandle和racks进行到同构的分类，并计算了有限循环群holomorphs的类数。
Notes on $B$-groups
本文通过区分 $B$-群与 $BS$-群，构造了无穷多个 $B$-群但非 $BS$-群的例子，揭示了 Schur 环方法在原始群研究中的局限性。
$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $Γ$ are groups of order $p^{2} q$ with non-is…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。