Ñåìèíàð 5. ÃÐÓÏÏÛ. ÐÅØÅÍÈÅ ÓÐÀÂÍÅÍÈÉ Â ÃÐÓÏÏÀÕ • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit 1. Ãðóïïû Îïðåäåëåíèå 5.1. Ýëåìåíò y ìíîæåñòâà G íàçûâàåòñÿ ëåâûì (ïðàâûì) îáðàòíûì ê ýëåìåíòó x îòíîñèòåëüíî äàííîé îïåðàöèè, åñëè y ∗ x = 1 ( x ∗ y = 1 ). Ýëåìåíò y , êîòîðûé ÿâëÿåòñÿ îäíîâðåìåííî ëåâûì è ïðàâûì îáðàòíûì, íàçûâàåòñÿ ïðîñòî îáðàòíûì ê x îòíîñèòåëüíî äàííîé îïåðàöèè. Îïðåäåëåíèå 5.2. Ìîíîèä íàçûâàåòñÿ ãðóïïîé, åñëè â íåì äëÿ êàæäîãî ýëåìåíòà ñóùåñòâóåò îáðàòíûé. Òåîðåìà 1.  ëþáîé ãðóïïå G = (G, ·) äëÿ êàæäîãî ýëåìåíòà a ∈ G ýëåìåíò, îáðàòíûé ê a , åäèíñòâåííûé. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit ×òîáû ïðîâåðèòü, ÷òî àëãåáðà (G, ∗) ÿâëÿåòñÿ ãðóïïîé, íóæíî 1) ïðîâåðèòü àññîöèàòèâíîñòü îïåðàöèè ∗ íà ìíîæåñòâå G; 2) íàéòè ýëåìåíò 1 ìíîæåñòâà G | íåéòðàëüíûé ýëåìåíò (åäèíèöó) îòíîñèòåëüíî îïåðàöèè ∗ ; 3) óáåäèòüñÿ, ÷òî äëÿ êàæäîãî ýëåìåíòà èç G ñóùåñòâóåò îáðàòíûé. Ãðóïïà íàçûâàåòñÿ êîììóòàòèâíîé (àáåëåâîé), åñëè åå îïåðàöèÿ êîììóòàòèâíà. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Ïðèìåð 1. Ðàññìîòðèì àëãåáðó (2A , 4, ∅) . Îïåðàöèÿ ñèììåòðè÷åñêîé ðàçíîñòè 1) àññîöèàòèâíà ( (A 4 B) 4 C = A 4(B 4 C)) ; 2) äëÿ ëþáîãî X ⊆ A X 4 ∅ = X , ò.å. ∅ | íåéòðàëüíûé ýëåìåíò îòíîñèòåëüíî äàííîé îïåðàöèè; 3) X 4 Y = ∅ òîãäà è òîëüêî òîãäà, êîãäà X = Y , ò.å. êàæäûé ýëåìåíò X ÿâëÿåòñÿ îáðàòíûì ñàì ê ñåáå. Ñëåäîâàòåëüíî, äàííàÿ àëãåáðà ÿâëÿåòñÿ ãðóïïîé. Ïîñêîëüêó îïåðàöèÿ 4 êîììóòàòèâíà A 4 B = B 4 A, òî äàííàÿ àëãåáðà ÿâëÿåòñÿ àáåëåâîé ãðóïïîé. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Çàäà÷à 6.1. Êàêèå èç óêàçàííûõ ìíîæåñòâ ñ îïåðàöèÿìè ÿâëÿþòñÿ ãðóïïàìè: (à) (N ∪ {0}, + ) ; (á) (Q, + ) ; (â) (R \ {0}, · ) . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Çàäà÷à 6.2. Êàêèå èç óêàçàííûõ ìíîæåñòâ êâàäðàòíûõ âåùåñòâåííûõ ìàòðèö îáðàçóþò ãðóïïó: (à) ìíîæåñòâî íåâûðîæäåííûõ ìàòðèö îòíîñèòåëüíî óìíîæåíèÿ? (á) ìíîæåñòâî íåâûðîæäåííûõ ìàòðèö îòíîñèòåëüíî ñëîæåíèÿ? (â) ìíîæåñòâî äèàãîíàëüíûõ ìàòðèö îäíîãî ïîðÿäêà (âêëþ÷àÿ íóëåâóþ) îòíîñèòåëüíî ñëîæåíèÿ? (ã) ìíîæåñòâî äèàãîíàëüíûõ ìàòðèö îäíîãî ïîðÿäêà, èñêëþ÷àÿ íóëåâóþ, îòíîñèòåëüíî óìíîæåíèÿ? Çàäà÷à 6.3. Ïóñòü M | íåêîòîðîå ìíîæåñòâî. ßâëÿåòñÿ ëè ãðóïïîé àëãåáðà (2M , ∩ ) ? • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit 2. Ðåøåíèå óðàâíåíèé â ãðóïïàõ Òåîðåìà 2.  ëþáîé ãðóïïå G ëþáîå óðàâíåíèå âèäà a · x = b èëè x · a = b èìååò åäèíñòâåííîå ðåøåíèå. Ðåøåíèå èìååò âèä: x = a−1 · b èëè x = b · a−1. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Ïðèìåð 2.  ãðóïïå S3 ðåøèì óðàâíåíèå 1 2 3 3 1 2 ◦X ◦ 1 2 3 2 3 1 = 1 2 3 3 2 1 . Óìíîæèì óðàâíåíèå ñëåâà íà 1 2 3 3 1 2 −1 = 1 2 3 2 3 1 , ïîëó÷èì: X◦ 1 2 3 2 3 1 = 1 2 3 2 1 3 . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Äàëåå, óìíîæàÿ ïîëó÷åííîå óðàâíåíèå ñïðàâà íà 1 2 3 2 3 1 −1 = 1 2 3 3 1 2 îêîí÷àòåëüíî ïîëó÷èì X= 1 2 3 1 2 3 1 2 3 · = = (2 3). 2 1 3 3 1 2 1 3 2 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Çàäà÷à 6.4. Ðåøèòü óðàâíåíèå â ãðóïïå S4 : 1 2 3 4 4 2 1 3 X 1 2 3 4 3 2 1 4 = (1 2) ; • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Çàäà÷à 6.5.  àääèòèâíîé ãðóïïå âû÷åòîâ ïî ìîäóëþ 5 Z⊕ 5 ðåøèòü óðàâíåíèå 4 ⊕5 x = 1 . Òàáëèöà Êýëè äëÿ ãðóïïû ({0, 1, 2, 3, 4}, ⊕5 ) : ⊕5 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Çàäà÷à 6.6.  ìóëüòèïëèêàòèâíîé ãðóïïå âû÷åòîâ ïî ìîäóëþ 5 Z 5 ðåøèòü óðàâíåíèå 4 5 x = 3 . Òàáëèöà Êýëè äëÿ ãðóïïû ({1, 2, 3, 4}, 5 ) : 5 1 2 3 4 1 1 2 3 4 2 2 4 1 3 3 3 1 4 2 4 4 3 2 1 • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Äîìàøíåå çàäàíèå Çàäà÷à Ä5.1. Ïóñòü A = {x, y, z} | ìíîæåñòâî áóêâ, à A∗ | ìíîæåñòâî âñåõ ñëîâ, êîòîðûå ìîæíî ñîñòàâèòü èç ýòèõ áóêâ ñ ïîâòîðåíèÿìè. Êîíêàòåíàöèåé äâóõ ñëîâ íàçûâàåòñÿ ñëîâî, ïîëó÷åííîå èõ ñêëåèâàíèåì\, íàïðèìåð: " xxy + yzxx = xxyyzxx . Ïóñòîå ñëîâî îáîçíà÷àþò λ . Ïîêàçàòü, ÷òî (A∗ , +) | ìîíîèä. Çàäà÷à Ä5.2 Ïóñòü M | íåêîòîðîå ìíîæåñòâî. ßâëÿåòñÿ ëè àëãåáðà (2M , ∪ ) ìîíîèäîì? ãðóïïîé? Çàäà÷à Ä5.3. Ðåøèòü óðàâíåíèå â ãðóïïå S4 : (1 2)(3 4)X(1 3) = 1 2 3 4 4 2 1 3 . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Çàäà÷à Ä5.4. Âûïèñàòü òàáëèöó Êýëè äëÿ ìíîæåñòâà ïîäñòàíîâîê {ε, (12)(34), (13)(24), (14)(23)} ñ îïåðàöèåé êîìïîçèöèè ïîäñòàíîâîê. Çàäà÷à Ä5.5.  àääèòèâíîé ãðóïïå âû÷åòîâ ïî ìîäóëþ 7 Z⊕7 ðåøèòü óðàâíåíèå 4 ⊕7 x = 2 . Çàäà÷à Ä5.6  ìóëüòèïëèêàòèâíîé ãðóïïå âû÷åòîâ ïî ìîäóëþ 7 Z 7 ðåøèòü óðàâíåíèå 6 5 x = 5 . Çàäà÷à Ä5.7  ìóëüòèïëèêàòèâíîé ãðóïïå âû÷åòîâ ïî ìîäóëþ 31 Z 31 ðåøèòü óðàâíåíèå 4 31 x = 5 . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit