Семинар 5. ГРУППЫ. РЕШЕНИЕ УРАВНЕНИЙ В ГРУППАХ

Ñåìèíàð 5. ÃÐÓÏÏÛ.
ÐÅØÅÍÈÅ ÓÐÀÂÍÅÍÈÉ
 ÃÐÓÏÏÀÕ
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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. Â ëþáîé ãðóïïå G = (G, ·) äëÿ êàæäîãî
ýëåìåíòà a ∈ G ýëåìåíò, îáðàòíûé ê a , åäèíñòâåííûé.
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×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.
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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.
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Çàäà÷à 6.1. Êàêèå èç óêàçàííûõ ìíîæåñòâ ñ îïåðàöèÿìè
ÿâëÿþòñÿ ãðóïïàìè:
(à) (N ∪ {0}, + ) ;
(á) (Q, + ) ;
(â) (R \ {0}, · ) .
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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 , ∩ ) ?
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2. Ðåøåíèå óðàâíåíèé â ãðóïïàõ
Òåîðåìà 2. Â 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.
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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
.
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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
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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) ;
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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
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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
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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
.
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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 .
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