Ciklichna grupa ce grupa yaka mozhe buti porodzhena odnim iz svoyih elementiv Tobto vsi elementi grupi ye stepenyami danogo elementa abo vikoristovuyuchi terminologiyu aditivnih grup vsi elementi grupi rivni n g displaystyle ng de g G n Z displaystyle g in G n in mathbb Z Formalno dlya multiplikativnih grup G a a n n Z displaystyle G langle a rangle left a n n in mathbb Z right dlya aditivnih G a n a n Z displaystyle G langle a rangle left na n in mathbb Z right Zmist 1 Prikladi 2 Vlastivosti 3 Teorema pro pidgrupi ciklichnoyi grupi 3 1 Dovedennya 4 Div takozh 5 DzherelaPrikladi RedaguvatiGrupa Z displaystyle mathbb Z nbsp cilih chisel z operaciyeyu dodavannya Dana grupa ye prikladom neskinchennoyi ciklichnoyi grupi Grupa Z n Z displaystyle mathbb Z n mathbb Z nbsp cilih chisel za modulem n displaystyle n nbsp z operaciyeyu dodavannya Dana grupa ye prikladom skinchennoyi ciklichnoyi grupi Grupa koreniv z n displaystyle n nbsp go stepenya z 1 displaystyle 1 nbsp v mnozhini kompleksnih chisel z operaciyeyu mnozhennya Vlastivosti RedaguvatiVsi ciklichni grupi ye abelevimi Ce viplivaye z asociativnosti grupi Kozhna skinchenna ciklichna grupa izomorfna grupi Z n Z displaystyle mathbb Z n mathbb Z nbsp a kozhna neskinchenna ciklichna grupa izomorfna grupi Z displaystyle mathbb Z nbsp Spravdi dlya neskinchennoyi grupi mozhna vzyati yak izomorfizm vidobrazhennya sho perevodit a k displaystyle a k nbsp v k displaystyle k nbsp Dlya skinchennoyi grupi poryadku n vikoristovuyemo analogichne vidobrazhennya i vrahovuyemo sho a n 1 displaystyle a n 1 nbsp ta n 0 mod n displaystyle n equiv 0 pmod n nbsp U neskinchennoyi ciklichnoyi grupi ye dva tvirni elementi 1 displaystyle 1 nbsp ta 1 displaystyle 1 nbsp dlya skinchennoyi grupi poryadku n displaystyle n nbsp yih kilkist rivna funkciyi Ejlera f n displaystyle varphi n nbsp tobto kilkosti chisel menshih vid n displaystyle n nbsp i vzayemno prostih z n displaystyle n nbsp Dlya skinchennoyi ciklichnoyi grupi element k displaystyle k nbsp ye tvirnim todi j lishe todi koli vin vzayemno prostij z n displaystyle n nbsp Todi isnuyut a b Z displaystyle a b in mathbb Z nbsp dlya yakih vikonuyetsya a k b n 1 displaystyle ak bn 1 nbsp tobto a k 1 mod n displaystyle ak equiv 1 pmod n nbsp Vidpovidno 2 a k 2 mod n displaystyle 2ak equiv 2 pmod n nbsp i tak dlya vsih elementiv Navpaki yaksho a k 1 mod n displaystyle ak equiv 1 pmod n nbsp to a k 1 displaystyle ak 1 nbsp dilitsya na n displaystyle n nbsp tobto rivne n b displaystyle nb nbsp dlya deyakogo cilogo b displaystyle b nbsp Todi a k b n 1 displaystyle ak bn 1 nbsp sho mozhlivo lishe dlya vzayemno prostih chisel Yaksho poryadok grupi proste chislo to taka grupa ciklichna Ye naslidkom teoremi Lagranzha Yaksho grupa ne maye vlasnih pidgrup to vona ciklichna i yiyi poryadok proste chislo Teorema pro pidgrupi ciklichnoyi grupi RedaguvatiKlyuchovim rezultatom dlya ciklichnih grup ye nastupna teorema Kozhna pidgrupa ciklichnoyi grupi ye ciklichnoyu Dovedennya Redaguvati Nehaj G displaystyle G nbsp ciklichna grupa i H displaystyle H nbsp yiyi pidgrupa Vvazhatimemo sho G displaystyle G nbsp i H displaystyle H nbsp ne ye trivialnimi tobto mayut bilshe odnogo elementa Nehaj g displaystyle g nbsp tvirnij element grupi G displaystyle G nbsp a n displaystyle n nbsp najmenshe dodatne cile chislo take sho g n H displaystyle g n in H nbsp Tverdzhennya H g n displaystyle H langle g n rangle nbsp g n H displaystyle langle g n rangle subseteq H nbsp a g n z Z a g n z displaystyle forall a in langle g n rangle exists z in mathbb Z mid a g n z nbsp g n H g n z H a H displaystyle g n in H Rightarrow g n z in H Rightarrow a in H nbsp Vidpovidno g n H displaystyle langle g n rangle subseteq H nbsp H g n displaystyle H subseteq langle g n rangle nbsp Nehaj h H displaystyle h in H nbsp h H h G x Z h g x displaystyle h in H Rightarrow h in G Rightarrow exists x in mathbb Z mid h g x nbsp Zgidno z algoritmom dilennya q r Z 0 r lt n x q n r displaystyle exists q r in mathbb Z mid 0 leq r lt n land x qn r nbsp h g x g q n r g q n g r g n q g r g r h g n q displaystyle h g x g qn r g qn g r g n q g r Rightarrow g r h g n q nbsp h g n H g r H displaystyle h g n in H Rightarrow g r in H nbsp Zvazhayuchi na vibir n displaystyle n nbsp i te sho 0 r lt n displaystyle 0 leq r lt n nbsp oderzhuyemo r 0 displaystyle r 0 nbsp r 0 h g n q g 0 g n q g n displaystyle r 0 Rightarrow h g n q g 0 g n q in langle g n rangle nbsp Vidpovidno H g n displaystyle H subseteq langle g n rangle nbsp Div takozh RedaguvatiTeoriya grup Modulna arifmetika Ciklichnij grafDzherela RedaguvatiKurosh A G Teoriya grupp 3 e izd Moskva Nauka 1967 648 s ISBN 5 8114 0616 9 ros Vinberg E B Kurs algebri 4 e izd Moskva MCNMO 2011 592 s ISBN 978 5 94057 685 3 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 Ciklichna grupa amp oldid 36918080
