28206: Difference between revisions
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Arătăm, mai departe, că <math>H_1 \cap H_2 = H_2 \cap H_3 = H_1 \cap H_3 = \left\{e\right\}</math>. | Arătăm, mai departe, că <math>H_1 \cap H_2 = H_2 \cap H_3 = H_1 \cap H_3 = \left\{e\right\}</math>. | ||
Presupunem că există <math>a \in H_1 \cap H_2 </math>, cu <math> a \ne e </math>. Dacă <math> H_1</math> are cel puțin trei elemente, alegem <math> x \in H_1 \setminus \left\{e,a\right\}</math>. Cum <math>xa^{-1} \in H_1^\ast</math> și <math>a \in H_2^\ast</math>, rezultă că <math>xa^{-1}a = x \in H_3</math>, deci <math>H_1 \setminus \left\{e,a\right\} \subset H_3</math>, așadar <math>H_1 \setminus \left\{a\right\} \subset H_3</math>. Subgrupul <math>\langle H_1 \setminus \{a\}\rangle</math> generat de <math>H_1 \setminus \{a\}</math> este un grup al lui <math>H_1</math>. Deoarece ordinul lui <math>\langle H_1 \setminus \{a\}\rangle</math> este cel puțin <math>2</math> și trebuie să dividă ordinul lui <math>H_1</math>, rezultă că <math>\langle H_1 \setminus \{a\}\rangle = H_1</math>. Cum <math>H_3</math> este subgrup al lui <math>G</math>, iar <math>H_1 \setminus \left\{a\right\} \subset H_3</math>, rezultă că și <math>\langle H_1 \setminus \{a\}\rangle = H_1</math> este inclus în <math>H_3</math>, deci <math>a\in H_1 \cap H_2 \cap H_3 = \left\{e\right\}</math>, fals. Așadar, <math>H_1</math> nu poate avea cel puțin trei elemente. Dacă <math>H_1 = \{e,a\}</math>, atunci <math>H_2</math> are cel puțin trei elemente, pentru că <math> H_2 \ne H_1 </math>, și, ca mai înainte, rezultă că <math> H_2 \subset H_3</math>, așadar | Presupunem că există <math>a \in H_1 \cap H_2 </math>, cu <math> a \ne e </math>. Dacă <math> H_1</math> are cel puțin trei elemente, alegem <math> x \in H_1 \setminus \left\{e,a\right\}</math>. Cum <math>xa^{-1} \in H_1^\ast</math> și <math>a \in H_2^\ast</math>, rezultă că <math>xa^{-1}a = x \in H_3</math>, deci <math>H_1 \setminus \left\{e,a\right\} \subset H_3</math>, așadar <math>H_1 \setminus \left\{a\right\} \subset H_3</math>. Subgrupul <math>\langle H_1 \setminus \{a\}\rangle</math> generat de <math>H_1 \setminus \{a\}</math> este un grup al lui <math>H_1</math>. Deoarece ordinul lui <math>\langle H_1 \setminus \{a\}\rangle</math> este cel puțin <math>2</math> și trebuie să dividă ordinul lui <math>H_1</math>, rezultă că <math>\langle H_1 \setminus \{a\}\rangle = H_1</math>. Cum <math>H_3</math> este subgrup al lui <math>G</math>, iar <math>H_1 \setminus \left\{a\right\} \subset H_3</math>, rezultă că și <math>\langle H_1 \setminus \{a\}\rangle = H_1</math> este inclus în <math>H_3</math>, deci <math>a\in H_1 \cap H_2 \cap H_3 = \left\{e\right\}</math>, fals. Așadar, <math>H_1</math> nu poate avea cel puțin trei elemente. Dacă <math>H_1 = \{e,a\}</math>, atunci <math>H_2</math> are cel puțin trei elemente, pentru că <math> H_2 \ne H_1 </math>, și, ca mai înainte, rezultă că <math> H_2 \subset H_3</math>, așadar <math>a\in H_1 \cap H_2 \cap H_3 = \left\{e\right\}</math>, fals. În consecință, <math>H_1 \cap H_2 = \{e\} </math>. La fel se arată că <math>H_2 \cap H_3 = H_1 \cap H_3 = \left\{e\right\}</math>. Fie <math>|H_1| = m</math>, <math>|H_2| = n</math>, <math>|H_3| = p</math>, cu <math>H_1 = \{a_1, a_2, \ldots, a_m\}</math> | ||
Revision as of 10:37, 3 January 2025
28206 (Dana Heuberger)
Fie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(G,\cdot\right)} un grup cu elementul neutru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} care conține subgrupurile proprii, distincte, finite , și , astfel încât pentru orice permutare și orice , , rezultă că .
- Arătați că subgrupurile , și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_3} au același număr de elemente.
- Dacă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = H_1 \cup H_2 \cup H_3} , arătați că grupul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} este de tip Klein.
Soluție.
a) Pentru orice subgrup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} a lui Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} , notăm Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^\ast = H \setminus \left\{e\right\}} .
Arătăm mai întâi că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \cap H_2 \cap H_3 = \left\{e\right\}} .
Presupunem că există Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in H_1 \cap H_2 \cap H_3} , cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ne e} . Din ipoteză, rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1^\ast \cdot H_2^\ast \subset H_3} , deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1^\ast \cdot H_2^\ast \cdot H_3^\ast \subset H_3 \cdot H_3^\ast = H_3} . Cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in H_2^\ast} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1} \in H_3^\ast} , rezută că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e = a \cdot a^{-1} \in H_2^\ast \cdot H_3^\ast } , deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1^\ast \subset H_1^\ast \cdot H_2^\ast \cdot H_3^\ast \subset H_3} , așadar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1^\ast \subset H_3 } , adică Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \subset H_3 } . În mod analog, se arată că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_3 \subset H_1 } . Rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 = H_3 } , ceea ce contrazice ipoteza. În consecință, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \cap H_2 \cap H_3 = \left\{e\right\}} .
Arătăm, mai departe, că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \cap H_2 = H_2 \cap H_3 = H_1 \cap H_3 = \left\{e\right\}} .
Presupunem că există Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in H_1 \cap H_2 } , cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ne e } . Dacă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1} are cel puțin trei elemente, alegem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in H_1 \setminus \left\{e,a\right\}} . Cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xa^{-1} \in H_1^\ast} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in H_2^\ast} , rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xa^{-1}a = x \in H_3} , deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \setminus \left\{e,a\right\} \subset H_3} , așadar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \setminus \left\{a\right\} \subset H_3} . Subgrupul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle H_1 \setminus \{a\}\rangle} generat de Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \setminus \{a\}} este un grup al lui Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1} . Deoarece ordinul lui Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle H_1 \setminus \{a\}\rangle} este cel puțin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} și trebuie să dividă ordinul lui Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1} , rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle H_1 \setminus \{a\}\rangle = H_1} . Cum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_3} este subgrup al lui Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} , iar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \setminus \left\{a\right\} \subset H_3} , rezultă că și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle H_1 \setminus \{a\}\rangle = H_1} este inclus în Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_3} , deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in H_1 \cap H_2 \cap H_3 = \left\{e\right\}} , fals. Așadar, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1} nu poate avea cel puțin trei elemente. Dacă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 = \{e,a\}} , atunci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_2} are cel puțin trei elemente, pentru că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_2 \ne H_1 } , și, ca mai înainte, rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_2 \subset H_3} , așadar Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\in H_1 \cap H_2 \cap H_3 = \left\{e\right\}} , fals. În consecință, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 \cap H_2 = \{e\} } . La fel se arată că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_2 \cap H_3 = H_1 \cap H_3 = \left\{e\right\}} . Fie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |H_1| = m} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |H_2| = n} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |H_3| = p} , cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_1 = \{a_1, a_2, \ldots, a_m\}}