Group and semigroup theory research group
TalTech priority area
Research classification (Frascati)
Head of the research group
Research group member
Doctoral students
Keyword
group theory
endomorphism semigroups
applications of algebra in cryptography and mathematical control theory
Overview
The research is focused on the study of theconnections between groups and their endomorphism semigroups, and the applications ofgroup theory.The aim is to describe some well-knownclasses of finite groups by their endomorphismsemigroups and to decide whether a group isdetermined by its endomorphism semigroupin the class of all groups or not. We started todescribe all small groups that are determined bytheir endomorphism semigroups. Further, if agroup G is not determined by its endomorphismsemigroup, then to provide the complete list ofnonisomorphic groups having the endomorhismsemigroup isomorphic to that of G. As thecomputational group theory and the softwareGAP (http://www.gap-system.org) are becomingmore popular among people working in appliedalgebra, we started to develop algorithms whatare able to decide automatically whether or nota given finite group is determined by its endomorphism semigroup.
Important results
All groups of order 32 that are determinedby their endomorphism monoid were described. A general reduction theory of algebraicmodules were studied (joint work withC. Ling and C. Porter at London Imperial College). As a result, all unit reducible real quadratic number fields weredescribed. Moreover, it was shown thatthe unit reducible real quadratic numberfields appear to be very rear among all realquadratic fields (the natural density ofunit reducible real quadratic fields amongall real quadratic number fields is 0).
Related department
Teadusgrupiga seotud publikatsioonid
- Rassõlkin, A., Vaimann, T., Org, P., Leibak, A., Gordon, R., Priidel, E. ADCS development for student CubeSat satellites - TalTech case study // Proceedings of the Estonian Academy of Sciences (2021) 3, p. 268-285 : ill.
https://doi.org/10.3176/proc.2021.3.06 - Leibak, A. On the number of factorizations of t mod N and the probability distribution of Diffie-Hellman secret keys for many users // Advances in mathematics of communications (2023) vol. 17, 4, p. 920-927.
https://doi.org/10.3934/amc.2021029 - Kotta, Ü., Belikov, J., Halas, M., Leibak, A. Degree of Dieudonne determinant defines the order of nonlinear system // International journal of control (2019) vol. 92, 3, p. 518-527.
https://doi.org/10.1080/00207179.2017.1361042 - Leibak, A. Michailichenko group of matrices over skew-fields // Математицки весник = Matematički Vesnik (2019) vol. 71 , 1-2, p. 155–161.
http://www.vesnik.math.rs/vol/mv191210.pdf