Page 298 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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GRAPHS AND GROUPS, GEOMETRIES AND GAP - G2G2 (MS-7)
[4] M. Erickson, S. Fernando, W. H. Haemers, D. Hardy, J. Hemmeter, Deza graphs: A gen-
eralization of strongly regular graphs, J. Combinatorial Design, 7 (1999) 359–405.
Recent progress in distance-regular graphs
Jack Koolen, koolen@ustc.edu.cn
University of Science and Technology of China, China
In this talk I report on recent progress in distance-regular graphs. This talk is based on joint
work with Meng Yue Cao (Beijing Normal University), Ying Ying Tan (Anhui Jianzhu Uni-
versity), Xiaoye Liang (Anhui University), Jongyook Park (Kyungpook National university),
Gary Greaves (Nanyang Technological University), Qianqian Yang (University of Science and
Technology of China), Zhi Qiao (Sichuan Normal University)
Recent results on pronormality of subgroups of odd index in finite groups
Natalia Maslova, butterson@mail.ru
Krasovskii Institute of Mathematics and Mechanics UB RAS, Russian Federation
We consider only finite groups. A subgroup H of a group G is said to be pronormal in G if H
and Hg are conjugate in H, Hg for each g ∈ G. Some of well-known examples of pronormal
subgroups are the following: normal subgroups; maximal subgroups; Sylow subgroups; Sylow
subgroups of proper normal subgroups; Hall subgroups of solvable groups. Some problems in
finite group theory, combinatorics, and permutation group theory were solved in terms of the
pronormality, see, for example [1, 11, 12].
In 2012, E. Vdovin and D. Revin [13] proved that the Hall subgroups (when they exist)
are pronormal in all simple groups and, guided by the analysis in their proof, they conjectured
that any subgroup of odd index of a simple group is pronormal in this group. This conjecture
was disproved in [5, 6]. In [4, 5, 6, 7], finite simple groups in which all the subgroups of odd
index are pronormal were studied. More detailed surveys of investigations on pronormality
of subgroups of odd index in finite (not necessary simple) groups could be found in survey
papers [3, 8]. These surveys contain new results and some conjectures and open problems.
One such open problem is to complete classification of finite simple groups in which all the
subgroups of odd index are pronormal. One more open problem involves the classification of
direct products of nonabelian simple groups in which the subgroups of odd index are pronormal.
A detailed motivation for these problems was provided in [2]. Note that there are examples of
nonabelian simple groups G such that all the subgroups of odd index are pronormal in G, but
the group G × G contains a non-pronormal subgroups of odd index (see [2, Proposition 1]).
In this talk, we discuss a recent progress in investigations on pronormality of subgroups of
odd index in finite groups. In particular, we have obtained the complete classification finite
simple exceptional groups of Lie type in which the subgroups of odd index are pronormal [9]
and have proved that the subgroups of odd index are pronormal in a direct product G of simple
symplectic groups over fields of odd characteristics if and only if the subgroups of odd index
are pronormal in each direct factor of G [10]; moreover, deciding the pronormality of a given
subgroup of odd index in the direct product of simple symplectic groups over fields of odd
characteristics is reducible to deciding the pronormality of some subgroup H of odd index in
296
[4] M. Erickson, S. Fernando, W. H. Haemers, D. Hardy, J. Hemmeter, Deza graphs: A gen-
eralization of strongly regular graphs, J. Combinatorial Design, 7 (1999) 359–405.
Recent progress in distance-regular graphs
Jack Koolen, koolen@ustc.edu.cn
University of Science and Technology of China, China
In this talk I report on recent progress in distance-regular graphs. This talk is based on joint
work with Meng Yue Cao (Beijing Normal University), Ying Ying Tan (Anhui Jianzhu Uni-
versity), Xiaoye Liang (Anhui University), Jongyook Park (Kyungpook National university),
Gary Greaves (Nanyang Technological University), Qianqian Yang (University of Science and
Technology of China), Zhi Qiao (Sichuan Normal University)
Recent results on pronormality of subgroups of odd index in finite groups
Natalia Maslova, butterson@mail.ru
Krasovskii Institute of Mathematics and Mechanics UB RAS, Russian Federation
We consider only finite groups. A subgroup H of a group G is said to be pronormal in G if H
and Hg are conjugate in H, Hg for each g ∈ G. Some of well-known examples of pronormal
subgroups are the following: normal subgroups; maximal subgroups; Sylow subgroups; Sylow
subgroups of proper normal subgroups; Hall subgroups of solvable groups. Some problems in
finite group theory, combinatorics, and permutation group theory were solved in terms of the
pronormality, see, for example [1, 11, 12].
In 2012, E. Vdovin and D. Revin [13] proved that the Hall subgroups (when they exist)
are pronormal in all simple groups and, guided by the analysis in their proof, they conjectured
that any subgroup of odd index of a simple group is pronormal in this group. This conjecture
was disproved in [5, 6]. In [4, 5, 6, 7], finite simple groups in which all the subgroups of odd
index are pronormal were studied. More detailed surveys of investigations on pronormality
of subgroups of odd index in finite (not necessary simple) groups could be found in survey
papers [3, 8]. These surveys contain new results and some conjectures and open problems.
One such open problem is to complete classification of finite simple groups in which all the
subgroups of odd index are pronormal. One more open problem involves the classification of
direct products of nonabelian simple groups in which the subgroups of odd index are pronormal.
A detailed motivation for these problems was provided in [2]. Note that there are examples of
nonabelian simple groups G such that all the subgroups of odd index are pronormal in G, but
the group G × G contains a non-pronormal subgroups of odd index (see [2, Proposition 1]).
In this talk, we discuss a recent progress in investigations on pronormality of subgroups of
odd index in finite groups. In particular, we have obtained the complete classification finite
simple exceptional groups of Lie type in which the subgroups of odd index are pronormal [9]
and have proved that the subgroups of odd index are pronormal in a direct product G of simple
symplectic groups over fields of odd characteristics if and only if the subgroups of odd index
are pronormal in each direct factor of G [10]; moreover, deciding the pronormality of a given
subgroup of odd index in the direct product of simple symplectic groups over fields of odd
characteristics is reducible to deciding the pronormality of some subgroup H of odd index in
296
