20.500.12556/DKUM-64705 Grupa kot direktni produkt svojih nerazcepnih podgrup Group as a direct product of its subgroups Cilj magistrskega dela je spoznati, pod katerimi pogoji lahko grupe razcepimo na direktni produkt nerazcepnih podgrup. Spoznamo nove pojme kot so notranji direktni produkt grup, razcepna grupa, padajoči in naraščujoči verižni pogoj. Ugotovimo, da če grupa G zadošča vsaj enemu izmed verižnih pogojev na svojih podgrupah edinkah, potem je grupa G enaka direktnemu produktu končnega števila nerazcepnih podgrup. Definiramo normalni endomorfizem in novo operacijo seštevanja preslikav. Ugotovimo, da je kompozitum naravnih vložitev in projekcij direktnega produkta grupe G, normalni endomorfizem grupe. S pomočjo lastnosti in povezav med vsemi novimi pojmi na koncu formuliramo in dokažemo Krull-Schmidtov izrek, ki pravi, da lahko vsako grupo G, ki zadošča pogojema naraščujočih in padajočih verig svojih podgrup edink, na enoličen način zapišemo v obliki direktnega produkta nerazcepnih podgrup grupe G. The aim of this dissertation is to present the conditions under which can groups be decomposed into direct product of its indecomposable subgroups. We meet with new terms, such as internal direct product of groups, indecomposable group, descending and ascending chain condition. We also find out that, if group G satisfies at least one of the chain conditions, then G is a direct product of a finite number of indecomposable subgroups. We define what a normal endomorphism and a new operation of adding mappings are. We also find out, that the compositum of canonical injections and projections of direct product of the group G is a normal endomorphism of the group G. With the help of properties and different connections between new terms we formulate and prove the Krull-Schmidt theorem, which says, that every group G, which has both chain conditions, can be uniquely written as a direct product of its indecomposable subgroups. grupa podgrupa notranji in zunanji direktni produkt grup nerazcepna grupa naraščujoči in padajoči verižni pogoj normalni nilpotentni endomorfizem Krull-Schmidtov izrek. Group Subgroup Internal and External Direct Product Indecomposable Group Ascending and Descending Chain Condition Normal Nilpotent Endomorphism Krull-Schmidt Theorem. true true false [B. Štulac] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2016-11-24 20:56:22 2017-01-19 15:00:32 2022-07-06 03:06:11 2019-07-11 14:10:36 2016 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512.54(043.2) 22866440 URN:SI:UM:DK:NPWUCD25 MAG_Stulac_Barbara_2016.pdf MAG_Stulac_Barbara_2016.pdf 1 C2F68A2094F9C4687218DFBCC9F37A0B 0f84f6575ac6b9a796399e85b22d97378a5e88669229bbcf4b0f9f8a29220d19 e0876451-7c0d-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=108479 Fakulteta za naravoslovje in matematiko 0 0 0