Page 292 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 292
GRAPHS AND GROUPS, GEOMETRIES AND GAP - G2G2 (MS-7)
The p-length of a p-solvable group and its character table
Neda Ahanjideh, ahanjidn@gmail.com
Shahrekord University, Islamic Republic of Iran
A question in the theory of finite p-solvable groups is to determine a bound for their p-lengths.
In several papers, it is shown that this bound can be read on the character table.
For a character χ of G, the number χc(1) = [G:kerχ] is called the co-degree of χ.
χ(1)
In this talk, we obtain a bound of the p-length of a p-solvable group by considering the
co-degrees of its irreducible characters.
On finitely generated quasi-scalar Jordan type algebras
Ilya Gorshkov, ilygor8@gmail.com
Sobolev Institute of Mathematics, Russian Federation
The concept of axial algebras was introduced by Hall, Rehren and Shpectorov. These algebras
are commutative, non-associative, and are generated by idempotents. We focus on the class of
axial algebras of Jordan type. It is well known that there is a unique bilinear form in algebras of
Jordan type such that any axis has length 1. We say that an algebra A with bilinear form is scalar
if the bilinear form defines an inner product on A. Quasi-scalarity is a useful generalization of
this concept. We say that an axial algebra A of Jordan type is quasi-scalar if for any two axes
a, b ∈ A the equality (a, b) = 1 holds if and only if a = b. It is easy to show that scalar
Jordan type algebras are quasi-scalar. We have studied the structure of quasi-scalar algebras. In
particular, it was proved that a finitely generated quasi-scalar algebra has a finite dimension if
and only if it is unital.
On maximal cliques in Paley graphs of square order
Sergey Goryainov, 44g@mail.ru
Chelyabinsk State University, Russian Federation
Coauthors: Alexander Masley, Leonid Shalaginov
In [1], Blokhuis studied maximum cliques in Paley graphs of square order P (q2). It was shown
that a clique of size q in P (q2) is necessarily a quadratic line in the corresponding affine plane
A(2, q).
Let r(q) denote the reminder after division of q by 4. In [2], for any odd prime power q, a
maximal (but not maximum) clique in P (q2) of size q+r(q) was constructed.
2
In [3], for any odd prime power q, a maximal clique in P (q2) of the same size q+r(q) was
2
constructed. This clique was shown to have a remarkable connection with eigenfunctions of
P (q2) that have minimum cardinality of support q + 1.
In this talk, we discuss the constructions of maximal cliques from [2] and [3] and establish
a correspondence between them.
Acknowledgments. Sergey Goryainov and Leonid Shalaginov are supported by RFBR accord-
ing to the research project 20-51-53023.
290
The p-length of a p-solvable group and its character table
Neda Ahanjideh, ahanjidn@gmail.com
Shahrekord University, Islamic Republic of Iran
A question in the theory of finite p-solvable groups is to determine a bound for their p-lengths.
In several papers, it is shown that this bound can be read on the character table.
For a character χ of G, the number χc(1) = [G:kerχ] is called the co-degree of χ.
χ(1)
In this talk, we obtain a bound of the p-length of a p-solvable group by considering the
co-degrees of its irreducible characters.
On finitely generated quasi-scalar Jordan type algebras
Ilya Gorshkov, ilygor8@gmail.com
Sobolev Institute of Mathematics, Russian Federation
The concept of axial algebras was introduced by Hall, Rehren and Shpectorov. These algebras
are commutative, non-associative, and are generated by idempotents. We focus on the class of
axial algebras of Jordan type. It is well known that there is a unique bilinear form in algebras of
Jordan type such that any axis has length 1. We say that an algebra A with bilinear form is scalar
if the bilinear form defines an inner product on A. Quasi-scalarity is a useful generalization of
this concept. We say that an axial algebra A of Jordan type is quasi-scalar if for any two axes
a, b ∈ A the equality (a, b) = 1 holds if and only if a = b. It is easy to show that scalar
Jordan type algebras are quasi-scalar. We have studied the structure of quasi-scalar algebras. In
particular, it was proved that a finitely generated quasi-scalar algebra has a finite dimension if
and only if it is unital.
On maximal cliques in Paley graphs of square order
Sergey Goryainov, 44g@mail.ru
Chelyabinsk State University, Russian Federation
Coauthors: Alexander Masley, Leonid Shalaginov
In [1], Blokhuis studied maximum cliques in Paley graphs of square order P (q2). It was shown
that a clique of size q in P (q2) is necessarily a quadratic line in the corresponding affine plane
A(2, q).
Let r(q) denote the reminder after division of q by 4. In [2], for any odd prime power q, a
maximal (but not maximum) clique in P (q2) of size q+r(q) was constructed.
2
In [3], for any odd prime power q, a maximal clique in P (q2) of the same size q+r(q) was
2
constructed. This clique was shown to have a remarkable connection with eigenfunctions of
P (q2) that have minimum cardinality of support q + 1.
In this talk, we discuss the constructions of maximal cliques from [2] and [3] and establish
a correspondence between them.
Acknowledgments. Sergey Goryainov and Leonid Shalaginov are supported by RFBR accord-
ing to the research project 20-51-53023.
290
