Zadannya grupi v matematici sposib viznachennya grupi za dopomogoyu mnozhini porodzhuvalnih elementiv S takih sho kozhen element grupi mozhe buti zapisanij cherez dobutok cih elementiv i mnozhini spivvidnoshen porodzhuvalnih elementiv R Yak pravilo take zadannya poznachayetsya tak S R displaystyle langle S mid R rangle Zadannya grupi ye duzhe kompaktnim i zruchnim sposobom viznachennya grupi prote iz zadannya grupi chasto vazhko vstanoviti navit najprostishi vlastivosti grupi zokrema chi ye grupa skinchennoyu komutativnoyu trivialnoyu i t d Osoblivo chasto zadannya grup vikoristovuyetsya v kombinatornij i geometrichnij teoriyi grup a takozh topologiyi Zmist 1 Formalne viznachennya 2 Vlastivosti 3 Prikladi 4 Skinchennoporodzheni i skinchennozadani grupi 5 Div takozh 6 DzherelaFormalne viznachennya RedaguvatiNehaj T deyaka mnozhina a lt S gt vilna grupa nad ciyeyu mnozhinoyu Nehaj teper R deyaka mnozhina sliv nad S tobto deyaka pidmnozhina lt S gt Poznachimo cherez N normalne zamikannya mnozhini R tobto minimalnu normalnu pidgrupu grupi lt S gt sho mistit vsi elementi R Viznachimo teper faktorgrupu S R S N displaystyle langle S mid R rangle langle S rangle N nbsp Elementi mnozhini S nazivayutsya porodzhuvalnimi generuvalnimi elementami a elementi R spivvidnoshennyami Yaksho deyaka grupa izomorfna do pobudovanoyi vishe grupi S R displaystyle langle S mid R rangle nbsp to kazhut sho cya grupa maye zadannya S R displaystyle langle S mid R rangle nbsp Yaksho r R displaystyle r in R nbsp deyakij element mnozhini spivvidnoshen to chasto pishut r 1 Takozh vikoristovuyetsya viraz x y de x y lt S gt displaystyle x y in lt S gt nbsp i y 1 x R displaystyle y 1 x in R nbsp Vlastivosti RedaguvatiDlya kozhnoyi grupi isnuye zadannyaSpravdi nehaj G deyaka grupa Poznachimo cherez lt G gt vilnu grupu nad mnozhinoyu elementiv G Todi zgidno z vlastivostyami vilnoyi grupi odinichne vidobrazhennya z G v G yedinim chinom prodovzhuyetsya do gomomorfizmu z lt G gt v G Poznachimo teper R mnozhinu elementiv lt G gt sho vhodyat do yadra cogo gomomorfizmu Todi lt G R gt ye odnim iz sposobiv zadannya grupi Zrozumilo sho ce zadannya ye duzhe nadlishkovim Teorema Dika Yaksho G S R displaystyle G langle S mid R rangle nbsp a H S R R displaystyle H langle S mid R cup R rangle nbsp tobto mnozhini porodzhuvalnih elementiv u dvoh grup odnakovi i mnozhina spivvidnoshen grupi H mistit vsi spivvidnoshennya grupi G i mozhlivo she j inshi todi H izomorfna deyakij faktorgrupi lt G gt Spravdi yaksho N normalne zamikannya R a N normalne zamikannya R R displaystyle R cup R nbsp todi N N displaystyle N subset N nbsp Todi zgidno z teoremoyu pro izomorfizm mayemo S N S N N N displaystyle langle S rangle N langle S rangle N N N nbsp sho j dovodit tverdzhennya Prikladi RedaguvatiV podanij nizhche tablici pokazani deyaki zadannya grup Dlya usih grup vibrani najprostishi zadannya Grupa Zadannya grupi KomentariVilna grupa na mnozhini S S displaystyle langle S mid varnothing rangle nbsp Grupa vilna bo nemaye spivvidnoshen Cn ciklichna grupa poryadku n a a n displaystyle langle a mid a n rangle nbsp D2n digedralna grupa poryadku 2n r f r n f 2 r f 2 displaystyle langle r f mid r n f 2 rf 2 rangle nbsp r povorot f simetrichne vidobrazhennyaD bezmezhna digedralna grupa r f f 2 r f 2 displaystyle langle r f mid f 2 rf 2 rangle nbsp Dicn diciklichna grupa r f r 2 n 1 r n f 2 f r f 1 r 1 displaystyle langle r f mid r 2n 1 r n f 2 frf 1 r 1 rangle nbsp Z Z x y x y y x displaystyle langle x y mid xy yx rangle nbsp Zm Zn x y x m 1 y n 1 x y y x displaystyle langle x y mid x m 1 y n 1 xy yx rangle nbsp Vilna abeleva grupa S S R displaystyle langle S mid R rangle nbsp de R mnozhina vsih komutatoriv elementiv SSimetrichna grupa Sn porodzhuvalni elementi s 1 s n 1 displaystyle sigma 1 ldots sigma n 1 nbsp spivvidnoshennya s i 2 1 displaystyle sigma i 2 1 nbsp s i s j s j s i if j i 1 displaystyle sigma i sigma j sigma j sigma i mbox if j neq i pm 1 nbsp s i s i 1 s i s i 1 s i s i 1 displaystyle sigma i sigma i 1 sigma i sigma i 1 sigma i sigma i 1 nbsp Tut s i displaystyle sigma i nbsp perestanovka sho minyaye miscyami i ij element z i 1 im the braid group Bn porodzhuvalni elementi s 1 s n 1 displaystyle sigma 1 ldots sigma n 1 nbsp spivvidnoshennya s i s j s j s i if j i 1 displaystyle sigma i sigma j sigma j sigma i mbox if j neq i pm 1 nbsp s i s i 1 s i s i 1 s i s i 1 displaystyle sigma i sigma i 1 sigma i sigma i 1 sigma i sigma i 1 nbsp Tetraedralna grupa T A4 s t s 2 t 3 s t 3 displaystyle langle s t mid s 2 t 3 st 3 rangle nbsp Oktaedralna grupa O S4 s t s 2 t 3 s t 4 displaystyle langle s t mid s 2 t 3 st 4 rangle nbsp Ikosaedralna grupa I A5 s t s 2 t 3 s t 5 displaystyle langle s t mid s 2 t 3 st 5 rangle nbsp Grupa kvaternioniv Q i j i 4 i 2 j 2 i j i j 1 displaystyle langle i j mid i 4 i 2 j 2 ijij 1 rangle nbsp S L 2 Z displaystyle SL 2 mathbb Z nbsp a b a b a b a b a b a 4 displaystyle langle a b mid aba bab aba 4 rangle nbsp G L 2 Z displaystyle GL 2 mathbb Z nbsp a b j a b a b a b a b a 4 j 2 j a 2 j b 2 displaystyle langle a b j mid aba bab aba 4 j 2 ja 2 jb 2 rangle nbsp P S L 2 Z displaystyle PSL 2 mathbb Z nbsp a b a 2 b 3 displaystyle langle a b mid a 2 b 3 rangle nbsp PSL2 Z ye vilnim dobutkom ciklichnih grup Z2 i Z3Grupa Gejzenberga x y z z x y x 1 y 1 x z z x y z z y displaystyle langle x y z mid z xyx 1 y 1 xz zx yz zy rangle nbsp Grupa Baumslaga Solitera B m n a b a n b a m b 1 displaystyle langle a b mid a n ba m b 1 rangle nbsp Grupa Titsa a 2 b 3 a b 13 a b 5 a b a b 4 a b a b a b a b a b 1 6 1 displaystyle a 2 b 3 ab 13 a b 5 a bab 4 ababababab 1 6 1 nbsp Skinchennoporodzheni i skinchennozadani grupi RedaguvatiYaksho dlya deyakoyi grupi isnuye zadannya zi skinchennoyu mnozhinoyu porodzhuvalnih elementiv to taka grupa nazivayetsya skinchennoporodzhenoyu Yaksho dlya deyakoyi grupi isnuye zadannya zi skinchennoyu mnozhinoyu porodzhuvalnih elementiv i skinchennoyu mnozhinoyu spivvidnoshen to taka grupa nazivayetsya skinchennozadanoyu Div takozh RedaguvatiPeretvorennya TitceDzherela RedaguvatiCoxeter H S M and Moser W O J 1980 Generators and Relations for Discrete Groups New York Springer Verlag ISBN 0 387 09212 9 Johnson D L 1990 Presentations of Groups Cambridge Cambridge University Press ISBN 0 521 37824 9 Kurosh A G Teoriya grupp 3 e izd Moskva Nauka 1967 648 s ISBN 5 8114 0616 9 ros Dzhozef Rotman en An Introduction to the Theory of Groups 4th Springer Graduate Texts in Mathematics 1994 532 s ISBN 978 0387942858 angl Otrimano z https uk wikipedia org w index php title Zadannya grupi amp oldid 39436193
