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A note on 𝑑-maximal 𝑝-groups 关于𝑑-maximal𝑝-groups的说明
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-07-04 DOI: 10.1515/jgth-2022-0071
Messab Aiech, H. Zekraoui, Y. Guerboussa
{"title":"A note on 𝑑-maximal 𝑝-groups","authors":"Messab Aiech, H. Zekraoui, Y. Guerboussa","doi":"10.1515/jgth-2022-0071","DOIUrl":"https://doi.org/10.1515/jgth-2022-0071","url":null,"abstract":"Abstract A finite 𝑝-group 𝐺 is said to be 𝑑-maximal if d ⁢ ( H ) < d ⁢ ( G ) d(H)<d(G) for every subgroup H < G H<G , where d ⁢ ( G ) d(G) denotes the minimal number of generators of 𝐺. A similar definition can be formulated when 𝐺 is acted on by some group 𝐴. In this paper, we extend earlier results of Kahn and Laffey to this more general setting and we answer a question of Berkovich on minimal non-metacyclic 𝑝-groups.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89059815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Class-two quotients of finite permutation groups 有限置换群的第二类商
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-06-29 DOI: 10.1515/jgth-2022-0214
H. Meng, Xiuyun Guo
{"title":"Class-two quotients of finite permutation groups","authors":"H. Meng, Xiuyun Guo","doi":"10.1515/jgth-2022-0214","DOIUrl":"https://doi.org/10.1515/jgth-2022-0214","url":null,"abstract":"Abstract Let 𝐺 be a permutation group on a finite set and let 𝑝 be a prime. In this paper, we prove that the largest class-two 𝑝-quotient of 𝐺 has order at most p n / p p^{n/p} (or 2 3 ⁢ n / 4 2^{3n/4} if p = 2 p=2 ), where 𝑛 is the number of points moved by a Sylow 𝑝-subgroup of 𝐺. Further, we describe the groups whose largest class-two 𝑝-quotients can reach such a bound. This extends earlier work of Kovács and Praeger from 1989.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74404493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
More on chiral polytopes of type {4, 4, …, 4} with~solvable automorphism groups 关于具有~可解自同构群的{4,4，…，4}型手性多面体的更多研究
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-06-27 DOI: 10.1515/jgth-2022-0124
Wei-Juan Zhang
{"title":"More on chiral polytopes of type {4, 4, …, 4} with~solvable automorphism groups","authors":"Wei-Juan Zhang","doi":"10.1515/jgth-2022-0124","DOIUrl":"https://doi.org/10.1515/jgth-2022-0124","url":null,"abstract":"Abstract In a 2021 paper, Conder et al. constructed two infinite families of chiral 4-polytopes of type { 4 , 4 , 4 } {4,4,4} with solvable automorphism groups. Here we present a general construction for chiral polytopes of type { 4 , 4 , … , 4 } {4,4,dots,4} with rank 4, 5 and 6, which are obtained as Boolean covers of the unique tight regular polytope of the same type.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84400440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Commutator endomorphisms of totally projective abelian 𝑝-groups 全射影阿贝尔的交换子自同态𝑝-groups
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-06-23 DOI: 10.1515/jgth-2022-0197
P. Keef
{"title":"Commutator endomorphisms of totally projective abelian 𝑝-groups","authors":"P. Keef","doi":"10.1515/jgth-2022-0197","DOIUrl":"https://doi.org/10.1515/jgth-2022-0197","url":null,"abstract":"Abstract For a primary abelian group 𝐺, Chekhlov and Danchev (2015) defined three variations of Kaplansky’s notion of full transitivity by restricting one’s attention to the subgroup, the subring and the unitary subring of the endomorphism ring of 𝐺 generated by the collection of all commutator endomorphisms. They posed the problem of describing exactly which totally projective groups exhibit these forms of full transitivity. This problem, and some closely related questions, are completely answered using the Ulm function of 𝐺.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76078260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On weak commutativity in 𝑝-groups 关于𝑝-groups的弱交换性
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-06-15 DOI: 10.1515/jgth-2022-0165
R. Bastos, E. de Melo, R. de Oliveira, C. Monetta
{"title":"On weak commutativity in 𝑝-groups","authors":"R. Bastos, E. de Melo, R. de Oliveira, C. Monetta","doi":"10.1515/jgth-2022-0165","DOIUrl":"https://doi.org/10.1515/jgth-2022-0165","url":null,"abstract":"Abstract The weak commutativity group χ ⁢ ( G ) chi(G) is generated by two isomorphic groups 𝐺 and G φ G^{varphi} subject to the relations [ g , g φ ] = 1 [g,g^{varphi}]=1 for all g ∈ G gin G . We present new bounds for the exponent of χ ⁢ ( G ) chi(G) and its sections, when 𝐺 is a finite 𝑝-group.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75465203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
5-Regular prime graphs of finite nonsolvable groups 有限不可解群的5-正则素图
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-06-08 DOI: 10.1515/jgth-2023-0041
Qinghong Guo, Weijun Liu, Lu Lu
{"title":"5-Regular prime graphs of finite nonsolvable groups","authors":"Qinghong Guo, Weijun Liu, Lu Lu","doi":"10.1515/jgth-2023-0041","DOIUrl":"https://doi.org/10.1515/jgth-2023-0041","url":null,"abstract":"Abstract The prime graph Δ ⁢ ( G ) Delta(G) of a finite group 𝐺 is a graph whose vertex set is the set of prime factors of the degrees of all irreducible complex characters of 𝐺, and two distinct primes 𝑝 and 𝑞 are joined by an edge if the product p ⁢ q pq divides some character degree of 𝐺. In 2014, Tong-Viet [H. P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra 397 (2014), 18–31] proposed the following conjecture. Let 𝐺 be a group and let k ≥ 5 kgeq 5 be odd. If the prime graph Δ ⁢ ( G ) Delta(G) is 𝑘-regular, then Δ ⁢ ( G ) Delta(G) is a complete graph of order k + 1 k+1 . In this paper, we show that if the prime graph Δ ⁢ ( G ) Delta(G) of a finite nonsolvable group 𝐺 is 5-regular, then Δ ⁢ ( G ) Delta(G) is isomorphic to the complete graph K 6 K_{6} or possibly the graph depicted in the first figure below. Moreover, if 𝐺 is an almost simple group, then Δ ⁢ ( G ) Delta(G) is isomorphic to the complete graph K 6 K_{6} .","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85929951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A characterization of the simple Ree groups 2𝐹4(𝑞2) by their character codegrees 简单稀土族2𝐹4(𝑞2)的特征码度表征
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-05-23 DOI: 10.1515/jgth-2022-0119
Yong Yang
{"title":"A characterization of the simple Ree groups 2𝐹4(𝑞2) by their character codegrees","authors":"Yong Yang","doi":"10.1515/jgth-2022-0119","DOIUrl":"https://doi.org/10.1515/jgth-2022-0119","url":null,"abstract":"Abstract The codegree of a character 𝜒 of a finite group 𝐺 is cod ⁡ ( χ ) := | G : ker ⁡ ( χ ) | χ ⁢ ( 1 ) . operatorname{cod}(chi):=frac{lvert G:ker(chi)rvert}{chi(1)}. We show that the set of codegrees of the Ree groups F 4 2 ⁢ ( q 2 ) {}^{2}F_{4}(q^{2}) ( q 2 = 2 2 ⁢ n + 1 q^{2}=2^{2n+1} , n ≥ 1 ngeq 1 ) determines the groups up to isomorphism.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90764878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hochschild cohomology of symmetric groups and generating functions 对称群的Hochschild上同调与生成函数
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-03-09 DOI: 10.1515/jgth-2022-0130
D. Benson, R. Kessar, M. Linckelmann
{"title":"Hochschild cohomology of symmetric groups and generating functions","authors":"D. Benson, R. Kessar, M. Linckelmann","doi":"10.1515/jgth-2022-0130","DOIUrl":"https://doi.org/10.1515/jgth-2022-0130","url":null,"abstract":"Abstract In this article, we compute the dimensions of the Hochschild cohomology of symmetric groups over prime fields in low degrees. This involves us in studying some partition identities and generating functions of the dimensions in any fixed degree of the Hochschild cohomology of symmetric groups. We show that the generating function of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups differs from that in degree 0 by a rational function.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88821265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On the converse of Gaschütz’ complement theorem 关于gasch<s:1>兹补定理的逆
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-03-01 DOI: 10.1515/jgth-2022-0178
Benjamin Sambale
{"title":"On the converse of Gaschütz’ complement theorem","authors":"Benjamin Sambale","doi":"10.1515/jgth-2022-0178","DOIUrl":"https://doi.org/10.1515/jgth-2022-0178","url":null,"abstract":"Abstract Let 𝑁 be a normal subgroup of a finite group 𝐺. Let N ≤ H ≤ G Nleq Hleq G such that 𝑁 has a complement in 𝐻 and ( | N | , | G : H | ) = 1 (lvert Nrvert,lvert G:Hrvert)=1 . If 𝑁 is abelian, a theorem of Gaschütz asserts that 𝑁 has a complement in 𝐺 as well. Brandis has asked whether the commutativity of 𝑁 can be replaced by some weaker property. We prove that 𝑁 has a complement in 𝐺 whenever all Sylow subgroups of 𝑁 are abelian. On the other hand, we construct counterexamples if Z ⁢ ( N ) ∩ N ′ ≠ 1 mathrm{Z}(N)cap N^{prime}neq 1 . For metabelian groups 𝑁, the condition Z ⁢ ( N ) ∩ N ′ = 1 mathrm{Z}(N)cap N^{prime}=1 implies the existence of complements. Finally, if 𝑁 is perfect and centerless, then Gaschütz’ theorem holds for 𝑁 if and only if Inn ⁢ ( N ) mathrm{Inn}(N) has a complement in Aut ⁢ ( N ) mathrm{Aut}(N) .","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79949194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Stiefel–Whitney classes of representations of SL(2, 𝑞) SL(2，𝑞)表示的Stiefel-Whitney类
IF 0.5 3区 数学
Journal of Group Theory Pub Date : 2023-02-28 DOI: 10.1515/jgth-2022-0164
Neha Malik, S. Spallone
{"title":"Stiefel–Whitney classes of representations of SL(2, 𝑞)","authors":"Neha Malik, S. Spallone","doi":"10.1515/jgth-2022-0164","DOIUrl":"https://doi.org/10.1515/jgth-2022-0164","url":null,"abstract":"Abstract We describe the Stiefel–Whitney classes (SWCs) of orthogonal representations 𝜋 of the finite special linear groups G = SL ⁡ ( 2 , F q ) G=operatorname{SL}(2,mathbb{F}_{q}) , in terms of character values of 𝜋. From this calculation, we can answer interesting questions about SWCs of 𝜋. For instance, we determine the subalgebra of H * ⁢ ( G , Z / 2 ⁢ Z ) H^{*}(G,mathbb{Z}/2mathbb{Z}) generated by the SWCs of orthogonal 𝜋, and we also determine which 𝜋 have non-trivial mod 2 Euler class.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86986136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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