20.500.12556/RUL-176601 Lastnost $({\rm T})$ grupe avtomorfizmov proste grupe ${\rm Aut}({\mathbb F}_5)$ magistrsko delo On property $({\rm T})$ for the automorphism group of the free group ${\rm Aut}({\mathbb F}_5)$ V nalogi obravnavamo dokaz dolgo odprte domneve iz teorije grup, da ima grupa avtomorfizmov proste grupe s petimi generatorji, označena z ${\rm Aut}({\mathbb F}_5)$, lastnost $({\rm T})$. Dokaz temelji na teoriji upodobitev ter vključuje računalnik za reševanje problemov semidefinitnega programiranja. Kasneje je bilo na podoben način dokazano, da imajo lastnost $({\rm T})$ tudi grupe ${\rm Aut}({\mathbb F}_5)$ za $n > 5$. Predstavimo tudi aplikacijo rezultata v obliki algoritma za generiranje naključnih elementov končnih grup. We present a proof of a long-standing open problem in group theory, namely whether the automorphism group of the free group on five generators denoted ${\rm Aut}({\mathbb F}_5)$ has property $({\rm T})$. The proof relies on representation theory and incorporates computer assistance through the use of semidefinite programming. It was later shown that the groups ${\rm Aut}({\mathbb F}_5)$ for $n > 5$ also possess property $({\rm T})$ using a similar approach. We also discuss an application of this result in the form of an algorithm for generating random elements of finite groups called the product replacement algorithm. lastnost (T) upodobitev prosta grupa avtomorfizem semidefinitno programiranje property (T) representation free group automorphism semidefinite programming true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2025-12-05 08:15:27 2025-12-05 08:15:30 2025-12-22 04:17:51 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 156756 260045059 21073.pdf 21073.pdf 1 0C94E58B2216D970F7A6AE088251DFCF 52e963b8b1b5669d221bc21356568027e1a7d6f15b5a25933cbb06a671c85544 cc4252c2-d1a9-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=223030 Fakulteta za matematiko in fiziko 0 0 0