Locally projective graphs in Mathieu–Conway–Monster series appear in thin–thick pairs. A possible thick extension of a thin locally projective graph associated with the fourth Janko group has been questioned for a while. Such an extension could lead, if not to a new sporadic simple group, to something equally exciting. This paper resolves this issue ultimately in the non-existence form confirming that the list of 26 sporadic simple groups, although mysterious, is now stable. The result in fact concludes the classification project of locally projective graphs, which has been running for some twenty years.
{"title":"The non-existence of a super-Janko group","authors":"Alexander A Ivanov","doi":"10.3336/gm.58.2.09","DOIUrl":"https://doi.org/10.3336/gm.58.2.09","url":null,"abstract":"Locally projective graphs in Mathieu–Conway–Monster series appear in thin–thick pairs. A possible thick extension of a thin locally projective graph associated with the fourth Janko group has been questioned for a while. Such an extension could lead, if not to a new sporadic simple group, to something equally exciting. This paper resolves this issue ultimately in the non-existence form confirming that the list of 26 sporadic simple groups, although mysterious, is now stable. The result in fact concludes the classification project of locally projective graphs, which has been running for some twenty years.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139153839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A (p)-group (G) with the property that its every nonabelian subgroup has a trivial centralizer (namely only its center) is called a (CZ)-group. In Berkovich's monograph (see [1]) the description of the structure of a (CZ)-group was posted as a research problem. Here we provide further progress on this topic based on results proved in [5]. In this paper we have described the structure of (CZ)-groups (G) that possess a nonabelian normal subgroup of order (p^4) which is contained in the Frattini subgroup (Phi(G).) We manage to prove that such a group of order (p^4) is unique and that the order of the entire group (G) is less than or equal to (p^7), (p) being a prime. Additionally, all such groups (G) are shown to be of a class less than maximal.
{"title":"(CZ)-groups with nonabelian normal subgroup of order (p^4)","authors":"Mario-Osvin Pavcevic, Kristijan Tabak","doi":"10.3336/gm.58.2.11","DOIUrl":"https://doi.org/10.3336/gm.58.2.11","url":null,"abstract":"A (p)-group (G) with the property that its every nonabelian subgroup has a trivial centralizer (namely only its center) is called a (CZ)-group. In Berkovich's monograph (see [1]) the description of the structure of a (CZ)-group was posted as a research problem. Here we provide further progress on this topic based on results proved in [5]. In this paper we have described the structure of (CZ)-groups (G) that possess a nonabelian normal subgroup of order (p^4) which is contained in the Frattini subgroup (Phi(G).) We manage to prove that such a group of order (p^4) is unique and that the order of the entire group (G) is less than or equal to (p^7), (p) being a prime. Additionally, all such groups (G) are shown to be of a class less than maximal.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139153129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1853, Steiner posed a number of combinatorial (tactical) problems, which eventually led to a large body of research on Steiner systems. However, solutions to Steiner's questions coincide with Steiner systems only for strengths two and three. For larger strengths, essentially only one class of solutions to Steiner's tactical problems is known, found by Bussey more than a century ago. In this paper, the relationships among Steiner systems, perfect binary one-error-correcting codes, and solutions to Steiner's tactical problem (Bussey systems) are discussed. For the latter, computational results are provided for at most 15 points.
{"title":"Bussey systems and Steiner's tactical problem","authors":"C. Colbourn, Donald L. Kreher, P. R. Ostergard","doi":"10.3336/gm.58.2.04","DOIUrl":"https://doi.org/10.3336/gm.58.2.04","url":null,"abstract":"In 1853, Steiner posed a number of combinatorial (tactical) problems, which eventually led to a large body of research on Steiner systems. However, solutions to Steiner's questions coincide with Steiner systems only for strengths two and three. For larger strengths, essentially only one class of solutions to Steiner's tactical problems is known, found by Bussey more than a century ago. In this paper, the relationships among Steiner systems, perfect binary one-error-correcting codes, and solutions to Steiner's tactical problem (Bussey systems) are discussed. For the latter, computational results are provided for at most 15 points.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139154165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let (G) be a finite group. Denote by (psi(G)) the sum (psi(G)=sum_{xin G}|x|,) where (|x|) denotes the order of the element (x), and by (o(G)) the average element orders, i.e. the quotient (o(G)=frac{psi(G)}{|G|}.) We prove that (o(G) = o(A_5)) if and only if (G simeq A_5), where (A_5) is the alternating group of degree (5).
{"title":"On groups with average element orders equal to the average order of the alternating group of degree (5)","authors":"Marcel Herzog, P. Longobardi, M. Maj","doi":"10.3336/gm.58.2.10","DOIUrl":"https://doi.org/10.3336/gm.58.2.10","url":null,"abstract":"Let (G) be a finite group. Denote by (psi(G)) the sum (psi(G)=sum_{xin G}|x|,) where (|x|) denotes the order of the element (x), and by (o(G)) the average element orders, i.e. the quotient (o(G)=frac{psi(G)}{|G|}.) We prove that (o(G) = o(A_5)) if and only if (G simeq A_5), where (A_5) is the alternating group of degree (5).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139153045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A design is called quasi-symmetric if it has only two block intersection numbers. Using a method based on orbit matrices, we classify quasi-symmetric (2)-((28,12,11)) designs with intersection numbers (4), (6), and an automorphism of order (5). There are exactly (31,696) such designs up to isomorphism.
{"title":"Quasi-symmetric (2)-((28,12,11)) designs with an automorphism of order (5)","authors":"Renata Vlahovic Kruc, Vedran Krčadinac","doi":"10.3336/gm.58.2.01","DOIUrl":"https://doi.org/10.3336/gm.58.2.01","url":null,"abstract":"A design is called quasi-symmetric if it has only two block intersection numbers. Using a method based on orbit matrices, we classify quasi-symmetric (2)-((28,12,11)) designs with intersection numbers (4), (6), and an automorphism of order (5). There are exactly (31,696) such designs up to isomorphism.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139153246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hadi Kharaghani, Thomas Pender, Caleb Van't Land, Vlad Zaitsev
Bush-type Butson Hadamard matrices are introduced. It is shown that a nonextendable set of mutually unbiased Butson Hadamard matrices is obtained by adding a specific Butson Hadamard matrix to a set of mutually unbiased Bush-type Butson Hadamard matrices. A class of symmetric Bush-type Butson Hadamard matrices over the group (G) of (n)-th roots of unity is introduced that is also valid over any subgroup of (G). The case of Bush-type Butson Hadamard matrices of even order will be discussed.
{"title":"Bush-type Butson Hadamard matrices","authors":"Hadi Kharaghani, Thomas Pender, Caleb Van't Land, Vlad Zaitsev","doi":"10.3336/gm.58.2.07","DOIUrl":"https://doi.org/10.3336/gm.58.2.07","url":null,"abstract":"Bush-type Butson Hadamard matrices are introduced. It is shown that a nonextendable set of mutually unbiased Butson Hadamard matrices is obtained by adding a specific Butson Hadamard matrix to a set of mutually unbiased Bush-type Butson Hadamard matrices. A class of symmetric Bush-type Butson Hadamard matrices over the group (G) of (n)-th roots of unity is introduced that is also valid over any subgroup of (G). The case of Bush-type Butson Hadamard matrices of even order will be discussed.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139154127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 2002, P. Gaborit introduced two constructions of self-dual codes using quadratic residues, so-called pure and bordered construction, as a generalization of the Pless symmetry codes. In this paper, we further study conditions under which the pure and the bordered construction using Paley designs and Paley graphs yield self-dual codes. Special attention is given to the binary and ternary codes. Further, we construct (t)-designs from supports of the codewords of a particular weight in the binary and ternary codes obtained.
{"title":"Block designs from self-dual codes obtained from Paley designs and Paley graphs","authors":"Dean Crnkovi c, Ana Grbac, Andrea v Svob","doi":"10.3336/gm.58.2.02","DOIUrl":"https://doi.org/10.3336/gm.58.2.02","url":null,"abstract":"In 2002, P. Gaborit introduced two constructions of self-dual codes using quadratic residues, so-called pure and bordered construction, as a generalization of the Pless symmetry codes. In this paper, we further study conditions under which the pure and the bordered construction using Paley designs and Paley graphs yield self-dual codes. Special attention is given to the binary and ternary codes. Further, we construct (t)-designs from supports of the codewords of a particular weight in the binary and ternary codes obtained.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139154110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Satoshi Yoshiara shows in [7] that the Veronesean dual hyperovals over ({mathbb F}_2) are of split type. So far there exists no published proof that a Veronesean dual hyperoval over any finite field of even characteristic is of split type. In this note we give a quick proof of this fact.
{"title":"Splitness of the Veronesean dual hyperovals: a quick proof","authors":"Ulrich Dempwolff","doi":"10.3336/gm.58.2.12","DOIUrl":"https://doi.org/10.3336/gm.58.2.12","url":null,"abstract":"Satoshi Yoshiara shows in [7] that the Veronesean dual hyperovals over ({mathbb F}_2) are of split type. So far there exists no published proof that a Veronesean dual hyperoval over any finite field of even characteristic is of split type. In this note we give a quick proof of this fact.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139153889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This corrigendum is written to correct an error in Theorem 1 in [1].
为纠正文献[1]中定理1中的一个错误，编写了本勘误表。
{"title":"Corrigendum to “A generalization of Iseki's formula”","authors":"P. Panzone, M. Ferrari, L. Piovan","doi":"10.3336/gm.58.1.11","DOIUrl":"https://doi.org/10.3336/gm.58.1.11","url":null,"abstract":"This corrigendum is written to correct an error in Theorem 1 in [1].","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82878846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the new notion of semi-parallel real hypersurface in the complex quadric (Q^{m}). Moreover, we give a nonexistence theorem for semi-parallel Hopf real hypersurfaces in the complex quadric (Q^{m}) for (m geq 3).
{"title":"Semi-parallel Hopf real hypersurfaces in the complex quadric","authors":"Hyunjin Lee, Y. Suh","doi":"10.3336/gm.58.1.08","DOIUrl":"https://doi.org/10.3336/gm.58.1.08","url":null,"abstract":"In this paper, we introduce the new notion of semi-parallel real hypersurface in the complex quadric (Q^{m}). Moreover, we give a nonexistence theorem for semi-parallel Hopf real hypersurfaces in the complex quadric (Q^{m}) for (m geq 3).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79067718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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