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Valorisation of environmental resources

Mathematics 1.1

group theory

endomorphism semigroups

applications of algebra in cryptography and mathematical control theory

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.

‚ 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).

Department of cybernetics