28206: Difference between revisions

Revision as of 10:13, 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 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} , 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} 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} , astfel încât pentru orice permutare $\sigma \in S_{3}$ și orice $a\in H_{\sigma \left(1\right)}\setminus \left\{e\right\}$ , $b\in H_{\sigma \left(2\right)}\setminus \left\{e\right\}$ , rezultă că $ab\in H_{\sigma \left(3\right)}$ .

Arătați că subgrupurile $H_{1}$ , $H_{2}$ și $H_{3}$ 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

Revision as of 10:06, 3 January 2025 view source Andrei.Horvat (talk \| contribs) 700 edits mNo edit summary ← Older edit		Revision as of 10:13, 3 January 2025 view source Andrei.Horvat (talk \| contribs) 700 edits No edit summary Newer edit →
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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		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