Abstract A finite 𝑝-group 𝐺 is said to be 𝑑-maximal if d ( H ) < d ( G ) d(H)
摘要对于每一子群H
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
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.
摘要对所有G∈G G in G，由两个同构群𝐺和G φ G^ {varphi}根据关系[G, G φ]=1 [G, G^ {varphi}]=1生成弱交换群χ≠(G) chi (G)。当𝐺是有限的𝑝-group时，我们给出了χ¹(G) chi (G)的指数及其截面的新边界。
{"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}
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} .
{"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}
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.
{"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}
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.
{"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}
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) .
摘要设无穷大群𝐺的正规子群。设N≤H≤G N leq H leq G使得在𝐻中存在一个补元并且(| N |， | G:H |)=1 (lvert N rvert, lvert G:H rvert)=1。如果抛掷是阿贝尔的，则抛掷的一个定理断言抛掷在𝐺中也有一个补。Brandis问过，是否可以用一些较弱的性质来代替二进制运算的交换性。证明了当所有的Sylow子群都是阿贝时，在𝐺中存在一个互补。另一方面，我们构造了Z≠(N)∩N '≠1 mathrm{Z} (N) cap N^ {prime}neq 1的反例。对于亚元群，条件Z≠(N)∩N ' =1 mathrm{Z} (N) cap N^ {prime} =1暗示了补的存在性。最后，如果操作端是完美且无中心的，那么对于操作端，当且仅当Inn (N) mathrm{Inn} (N)在Aut (N) mathrm{Aut} (N)中有补时，gasch兹定理成立。
{"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}
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.
摘要描述了有限特殊线性群G= SL (2, F q) G=operatorname{SL}(2，mathbb{F}_{q})的正交表示的Stiefel-Whitney类(SWCs)。从这个计算中，我们可以回答一些关于量子力学的有趣问题。例如，我们确定了正交SWCs生成的H * * (G, Z /2) H^{*}(G，mathbb{Z}/2mathbb{Z})的子代数，并确定了哪些是非平凡模2欧拉类。
{"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}
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