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| --- | ||
| created: 2024-11-26T14:33:42 | ||
| up: "[[📖NI-MPI]]" | ||
| --- | ||
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| TARGET DECK: NI-MPI | ||
| FILE TAGS: NI-MPI cviceni10 status-toReview | ||
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| ### Cvičení 17.2 | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \left(Q \setminus \set{0} , \cdot \right) \\ | ||
| & N = \set{2k, k \in \mathbb{Z} , k \neq 0} \\ | ||
| & \text{\small{1)}} \\ | ||
| & N \subset M \\ | ||
| & \text{\small Je grupa?} (N,\cdot ) \\ | ||
| & 1 \notin N \Rightarrow \text{\small{Nejedná se o podgrupu}} \\ | ||
| \end{align*} | ||
| $$ | ||
|
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||
| ### Cvičení 17.2b | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & N = \set{2k+1, k \in \mathbb{Z}} \\ | ||
| & \text{\small{1) }} N \subset M \\ | ||
| & \text{\small{2) }} (N,\cdot ) \\ | ||
| & \text{\small{Chybí in verze vzhledem k násobení, není podgrupa}} | ||
| \end{align*} | ||
| $$ | ||
|
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||
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| ### Cvičení 17.2c | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & N = \set{2^k, n \in \mathbb{Z}} \\ | ||
| & \text{\small{1) }} N \subset M \\ | ||
| & \text{\small{2) }} (N, \cdot ) \\ | ||
| & \text{\small{a) }} a,b \in N, a \cdot b = 2^{m} \cdot 2^{m} = 2^{m+n} \in N \\ | ||
| & \text{\small{b) asoc.}} \\ | ||
| & \text{\small{c) }} 1 = 2^{0} \in N \\ | ||
| & \text{\small{d) }} a=2^{m} \quad q^{-1} = 2^{-m} \in N \\ | ||
| & a^{-1} \cdot a = 2^{-m} \cdot 2^{m} =2^{0} =1=e \\ | ||
| & \text{\small{e) komutativní}} \\ | ||
| & \Rightarrow (N,\cdot ) \text{\small{ je podgrupou}} | ||
| \end{align*} | ||
| $$ | ||
|
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||
| ### Cvičení 17.2d | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & N=\set{2^{n}\cdot 3^{m} ; n,m \in \mathbb{Z} } \\ | ||
| & \text{\small{1) }} N \subset M \\ | ||
| & \text{\small{2) }} \\ | ||
| & \text{\small{a) }} \\ | ||
| & a = 2^{n}3^{m} \\ | ||
| & b = 2^{k} 3^{l} \\ | ||
| & a \circ b = 2^{n}3^{m}2^{k} 3^{l}=2^{n+k}3^{m+l} \in N \\ | ||
| & \text{\small{b) asociativnost platí}} \\ | ||
| & \text{\small{c) }} 1 = 2^{0}3^{0} \in N \\ | ||
| & \text{\small{d) }} a^{-1}=2^{-n}3^{-m} \\ | ||
| & \Rightarrow (N, \cdot ) \text{\small{ je podgrupou}} | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 17.2e | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & N = \set{\tfrac{1+2n}{1+2m} : n,m \in \mathbb{Z} } \\ | ||
| & \text{\small{1) }} N \subset M \\ | ||
| & \text{\small{2) }} \\ | ||
| & \dots \\ | ||
| & \Rightarrow \text{\small{Je podgrupa}} | ||
| \end{align*} | ||
| $$ | ||
| ### Cvičení 17.3 | ||
|
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||
| | | a | b | c | d | | ||
| | --- | --- | --- | --- | --- | | ||
| | a | a | b | c | d | | ||
| | b | b | a | d | c | | ||
| | c | c | d | a | b | | ||
| | d | d | c | b | a | | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \\ | ||
| & \text{Triviální podgrupy} \\ | ||
| & \text{\small{řád 4: }} \set{a,b,c,d} \checkmark \\ | ||
| & \text{\small{řád 1: }} \set{a} \checkmark \\ | ||
| & \\ | ||
| & \text{Další podgrupy} \\ | ||
| & \text{\small{řád 2: }} \set{a, ?} \\ | ||
| & \set{a,b} \checkmark \\ | ||
| & \set{a,c} \checkmark \\ | ||
| & \set{a,d} \checkmark \\ | ||
| & \text{\small{řád 3: }} \set{a, ?, ?} \\ | ||
| & \text{\small{Žádnou podgrupu nenajdem}} \\ | ||
| \end{align*} | ||
| $$ | ||
| ### Cvičení 17.4 | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & G = \set{a+b\sqrt{2} ; a,b \in \mathbb{Q} , q \neq 0 \lor b \neq 0} \\ | ||
| & \text{\small{Je }} (G,\cdot ) \text{\small{ podgrupou }} (R \setminus \set{0} ; \cdot ) \text{\small{ ?}} \\ | ||
| & \text{\small{1) }} N \subset M \\ | ||
| & \text{\small{2) Vlastnosti grupy:}} \\ | ||
| & \text{\small{a) uzavřenost}} \\ | ||
| & m = a+b\sqrt{2} \\ | ||
| & n=c+d\sqrt{2} \\ | ||
| & m\cdot n = (a+b\sqrt{2})(c+d\sqrt{2}) = \\ | ||
| & = (ac+2bd)+(bc+ad)\sqrt{2} \in \mathbb{Q} \\ | ||
| & \text{\small{Pozor, musíme ověřit abych nedostal }} 0 \\ | ||
| & \text{\small{Víme z definice G že to jsou nenulová čísla a násobením nenulových reálných čísel nemohu dostat nulu}} \\ | ||
| & \text{\small{b) asociativita }} \checkmark \\ | ||
| & \text{\small{c) neutrální prvek }} \\ | ||
| & 1=1+0\sqrt{2} \\ | ||
| & \text{\small{d) inverze}} \\ | ||
| & m^{-1} = \tfrac{1}{a+b\sqrt{2} } \cdot \tfrac{a-b\sqrt{2} }{a-b\sqrt{2}} = \tfrac{a}{a^2 -2b^2 } -\tfrac{b}{a^2 -2b^2 } \sqrt{2} \in \mathbb{G} \\ | ||
| & \Rightarrow (G, \cdot ) \text{\small{ je podgrupou }} (R \setminus \set{0} ; \cdot ) | ||
| & \text{\small{e) uzavřenost}} \\ | ||
| \end{align*} | ||
| $$ | ||
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| ### Cvičení 17.5 | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & (M, \circ ) \\ | ||
| & G = \set{q \in M : a \text{\small{ je invertibilní}} } \\ | ||
| & \text{\small{Je }} (G, \circ ) \text{\small{ grupa?}} \\ | ||
| & a \in G \land b \in G \xrightarrow{?} a \circ b \in G \\ | ||
| & \\ | ||
| & \text{\small{kandidát }} b^{-1} \circ a^{-1} \\ | ||
| & (b^{-1} \circ a^{-1})\circ (a \circ b) = \\ | ||
| & = b^{-1} \circ (a^{-1}\circ a ) \circ b = \\ | ||
| & =b^{-1} \circ e \circ b= b^{-1} \circ b \circ e \\ | ||
| & (a \circ b) \circ (b^{-1} \circ a^{-1}) = a \circ a^{-1} =e \\ | ||
| & \text{\small{oba prvky }} (a \circ b), (b^{-1} \circ a^{-1}) \text{\small{ leží v G }} \\ | ||
| & \\ | ||
| & \text{Ověření vlastností grupy} \\ | ||
| & \text{\small{a) uzavřenost }} \checkmark \\ | ||
| & \text{\small{b) asociativita }} \checkmark \\ | ||
| & \text{\small{a) neutrální prvek }} \checkmark \\ | ||
| & \text{\small{a) inverzní prvek }} \checkmark \\ | ||
| & \Rightarrow (G,\circ ) \text{\small{ je grupa.}} \\ | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 18.1 | ||
|  | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \mathbb{Z}_{15}^{+} = (M,+_{15}) \\ | ||
| & M = \set{0,1,2,\dots 14} \\ | ||
| & \text{Generátory:} \\ | ||
| & g=1 \checkmark \#<g> =\# \mathbb{Z}_{15}^{+} = 15 \\ | ||
| & g = 14 \checkmark \\ | ||
| & \text{\small{Dle Lagrangeovy věty řád podgrupy musí být dělitel řádu celé grupy. }} \\ | ||
| & \text{\small{Mohou existovat jen podgrupy řádů:}} \\ | ||
| & 1,3,5,15 \\ | ||
| & \text{\small{řád 3: }} \set{0,5,10} \\ | ||
| & \text{\small{řád 5: }} \set{0,3,6,9,12} \\ | ||
| & \text{\small{Generátory jsou:}} \\ | ||
| & 1,2,4,7,8,11,13,14 \\ \\ | ||
| & \text{\small{Ověření kolik jich má být:}} \\ | ||
| & \varphi(15) = \varphi(5) \varphi(3) = (5^1 -5^{0})(3^{1}-3^{0})=4\cdot 2=8 \\ | ||
| & \\ | ||
| & \text{Inverzní prvky:} \\ | ||
| & a^{-1}=15-a\\ | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 18.2 | ||
|  | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \text{\small{řády }} 3, 5, 7 \\ | ||
| & 3\cdot 5\cdot 7 = 105 \\ | ||
| & \mathbb{Z}_{105}^{+} \\ | ||
| & \text{\small{podgrupa řádu 3: }} \set{0,35,70} \\ | ||
| & \text{\small{podgrupa řádu 5: }} \set{0,21,42,63,84} \\ | ||
| & \text{\small{podgrupa řádu 7: }} \set{0,15,30,45,60,75,90} \\ | ||
| \end{align*} | ||
| $$ | ||
| ### Cvičení 18.3a | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \left(\mathbb{Z,+} \right) \quad \text{\small{množina }} \set{2} \\ | ||
| & \\ | ||
| & \text{Řešení} \\ | ||
| & (M, +) \quad M=\set{2k, k \in \mathbb{Z}} | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 18.3b | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \left(\mathbb{Z,+} \right) \quad \text{\small{množina }} \set{2,3} \\ | ||
| & \\ | ||
| & \text{Řešení} \\ | ||
| & 2^{-1} \circ 3^{1} =1 \quad \text{\small{je generátor }} \left(\mathbb{Z,+} \right)\\ | ||
| & M = \mathbb{Z} | ||
| \end{align*} | ||
| $$ | ||
|
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| ### Cvičení 18.3c | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \left(\mathbb{Z,+} \right) \quad \text{\small{množina }} \set{2,5} \\ | ||
| & \\ | ||
| & \text{Řešení} \\ | ||
| & 2^{-2} \circ 5^{1} =1 \quad \text{\small{je generátor }} \left(\mathbb{Z,+} \right)\\ | ||
| & M = \mathbb{Z} | ||
| \end{align*} | ||
| $$ | ||
| ### Cvičení 18.3d | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \left(\mathbb{Z,+} \right) \quad \text{\small{množina }} \set{6,15} \\ | ||
| & \\ | ||
| & \text{Řešení} \\ | ||
| & \text{\small{nevyrobím }} 1 \text{\small{ ani }} -1 \\ | ||
| & 6^{-2} \circ 15^{1} =3=\gcd{(6,15)} \\ | ||
| & M = \set{3k, k \in \mathbb{Z}} | ||
| \end{align*} | ||
| $$ | ||
| ### Cvičení 18.3e | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \left(\mathbb{Z,+} \right) \quad \text{\small{množina }} \set{n,m}, n \neq m \\ | ||
| & \\ | ||
| & \text{Řešení} \\ | ||
| & N=\set{a \cdot k, k \in \mathbb{Z}, a = \gcd{(n,m)}} \\ | ||
| \end{align*} | ||
| $$ | ||
| ### Cvičení 18.4 | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \left(\mathbb{Z},+ \right) \\ | ||
| & N = \set{a\cdot k, a \in \mathbb{Z}, k \in \mathbb{Z}} \quad \text{\small{Řešení}} \\ | ||
| & | ||
| \end{align*} | ||
| $$ | ||
| ### Cvičení 18.5 | ||
|  | ||
|  | ||
| <!-- Latex Equation --> | ||
| $$ | ||
| \begin{align*} | ||
| & \mathbb{Z}_{11}^{\times} = (M, \times_{11} ) \\ | ||
| & M = \set{1,2, 3,4,5,6,7,8,9,10} \\ | ||
| & \# \mathbb{Z}_{11}^{\times}=10 \\ | ||
| & \\ | ||
| & \text{Generátory} \\ | ||
| & \text{\small{2 je generátor:}} \\ | ||
| & 2^2 = 4 \\ | ||
| & 2^3 = \\ | ||
| & \dots \\ | ||
| & \begin{rcases} 2^2 = 4 \\2^{5} = 2^{2} \cdot 2^{2} \cdot 2 = 4\cdot 4\cdot 2=10 \\ \end{rcases} \text{\small{ 2 je generátor}} \\ | ||
| & \dots \\ | ||
| & 2^{10} = 1 \\ | ||
| & \text{\small{možné řády podgrup: }} 2, 5 \\ | ||
| & \text{\small{Chceme selektivně vybrat ty mocniny}} \\ | ||
| & \text{\small{Hledáme čísla nesoudělná s 10: }} \\ | ||
| & 1,3,7,9 \\ | ||
| & \text{\small{Generátory:}} \\ | ||
| & 2^{1} = 2 \\ | ||
| & 2^{3} = 8 \\ | ||
| & 2^{7} = 7 \\ | ||
| & 2^{9} = 6 \\ | ||
| & \\ | ||
| & \text{Inverzní prvky:} \\ | ||
| & 1 \rightarrow 1 \\ | ||
| & 2 \rightarrow 6 \\ | ||
| & 3 \rightarrow 4 \\ | ||
| & 4 \rightarrow 3 \\ | ||
| & 5 \rightarrow 9 \\ | ||
| & 6 \rightarrow 2 \\ | ||
| & 7 \rightarrow 8 \\ | ||
| & 8 \rightarrow 7 \\ | ||
| & 9 \rightarrow 5 \\ | ||
| & 10 \rightarrow 10 \\ | ||
| \end{align*} | ||
| $$ |