Lemat Goursata – twierdzenie teorii grup charakteryzujące podgrupy iloczynu prostego dwóch grup .
Pierwszy raz pojawiło się ono w pracy Édouarda Goursata pt. Sur les substitutions orthogonales et les divisions régulières de l’espace („O podstawieniach ortogonalnych i podziałach regularnych przestrzeni”) z 1889 roku[1] osobnej sekcji pokazane zostanie, w jaki sposób można udowodnić za jego pomocą lemat Zassenhausa .
W teorii grup dostępne są trzy standardowe sposoby konstruowania nowych grup z istniejących:
wzięcie podgrupy H {\displaystyle H} G , {\displaystyle G,} wzięcie ilorazu G / H {\displaystyle G/H} H {\displaystyle H} podgrupą normalną ) oraz wzięcie iloczynu prostego G 1 × G 2 {\displaystyle G_{1}\times G_{2}} G 1 {\displaystyle G_{1}} G 2 . {\displaystyle G_{2}.} Dla każdej z tych konstrukcji można zapytać: jak wyglądają podgrupy uzyskanej grupy? W dwóch pierwszych przypadkach odpowiedź jest prosta: podgrupa L {\displaystyle L} H {\displaystyle H} L {\displaystyle L} G {\displaystyle G} H {\displaystyle H} wniosku z twierdzenia o odpowiedniości wynika, że podgrupy G / H {\displaystyle G/H} J / H , {\displaystyle J/H,} J {\displaystyle J} G , {\displaystyle G,} H ⊆ J ⊆ G {\displaystyle H\subseteq J\subseteq G} J / H ⊴ G / H {\displaystyle J/H\trianglelefteq G/H} J ⊴ G {\displaystyle J\trianglelefteq G} G 1 {\displaystyle G_{1}} G 2 {\displaystyle G_{2}} G 1 × G 2 . {\displaystyle G_{1}\times G_{2}.}
Iloczyn prosty G 1 × G 2 {\displaystyle G_{1}\times G_{2}} G 1 {\displaystyle G_{1}} G 2 {\displaystyle G_{2}} { ( g 1 , g 2 ) : g i ∈ G i } {\displaystyle \{(g_{1},g_{2})\colon g_{i}\in G_{i}\}} mnożeniem określonym po współrzędnych : ( g 1 , g 2 ) ( h 1 , h 2 ) = ( g 1 h 1 , g 2 h 2 ) ; {\displaystyle (g_{1},g_{2})(h_{1},h_{2})=(g_{1}h_{1},g_{2}h_{2});} elementem neutralnym jest ( e 1 , e 2 ) , {\displaystyle (e_{1},e_{2}),} element odwrotny to ( g 1 , g 2 ) − 1 = ( g 1 − 1 , g 2 − 1 ) . {\displaystyle (g_{1},g_{2})^{-1}=(g_{1}^{-1},g_{2}^{-1}).} H i {\displaystyle H_{i}} G i , {\displaystyle G_{i},} H 1 × H 2 {\displaystyle H_{1}\times H_{2}} G 1 × G 2 {\displaystyle G_{1}\times G_{2}} [a] podiloczynem ; więcej H 1 × H 2 {\displaystyle H_{1}\times H_{2}} G 1 × G 2 {\displaystyle G_{1}\times G_{2}} H i ⊴ G i {\displaystyle H_{i}\trianglelefteq G_{i}} [b]
Jako wprowadzenie przedstawione zostanie rozwiązanie następującego problemu:
które pary grup G 1 {\displaystyle G_{1}} G 2 {\displaystyle G_{2}} G 1 × G 2 {\displaystyle G_{1}\times G_{2}} G 1 × G 2 {\displaystyle G_{1}\times G_{2}} Odpowiedź daje następujące
Stwierdzenie Niech G 1 {\displaystyle G_{1}} G 2 {\displaystyle G_{2}} nietrywialnymi grupami. Każda podgrupa w G 1 × G 2 {\displaystyle G_{1}\times G_{2}} g i ∈ G i {\displaystyle g_{i}\in G_{i}} względnie pierwsze rzędy [c] wykorzystujące poniższy
Lemat Niech G 1 {\displaystyle G_{1}} G 2 {\displaystyle G_{2}} nietrywialnymi grupami. Wówczas G 1 × G 2 {\displaystyle G_{1}\times G_{2}} cykliczna wtedy i tylko wtedy, gdy G 1 {\displaystyle G_{1}} G 2 {\displaystyle G_{2}} względnie pierwszych rzędach [d] Niech G 1 , G 2 {\displaystyle G_{1},G_{2}} grupami .
Niech H {\displaystyle H} podgrupą w G 1 × G 2 . {\displaystyle G_{1}\times G_{2}.} H 11 = { a ∈ G 1 : ( a , e 2 ) ∈ H } , {\displaystyle H_{11}=\{a\in G_{1}\colon (a,e_{2})\in H\},} H 21 = { b ∈ G 2 : ( e 1 , b ) ∈ H } {\displaystyle H_{21}=\{b\in G_{2}\colon (e_{1},b)\in H\}} H 12 = { a ∈ G 1 : ( a , b ) ∈ H dla pewnego b ∈ G 2 } , {\displaystyle H_{12}=\{a\in G_{1}\colon (a,b)\in H{\text{ dla pewnego }}b\in G_{2}\},} H 22 = { b ∈ G 2 : ( a , b ) ∈ H dla pewnego a ∈ G 1 } . {\displaystyle H_{22}=\{b\in G_{2}\colon (a,b)\in H{\text{ dla pewnego }}a\in G_{1}\}.} Wówczas H i 1 ⊆ H i 2 {\displaystyle H_{i1}\subseteq H_{i2}} G i , {\displaystyle G_{i},} H i 1 ⊴ H i 2 , {\displaystyle H_{i1}\trianglelefteq H_{i2},} φ H : H 12 / H 11 → H 22 / H 21 {\displaystyle \varphi _{H}\colon H_{12}/H_{11}\to H_{22}/H_{21}} φ H ( a H 11 ) = b H 21 , {\displaystyle \varphi _{H}(aH_{11})=bH_{21},} ( a , b ) ∈ H , {\displaystyle (a,b)\in H,} Co więcej: jeśli H ⊴ G 1 × G 2 , {\displaystyle H\trianglelefteq G_{1}\times G_{2},} H i 1 , H i 2 ◃ G i {\displaystyle H_{i1},H_{i2}\triangleleft G_{i}} H i 2 / H i 1 ⊆ Z ( G i / H i 1 ) , {\displaystyle H_{i2}/H_{i1}\subseteq \mathrm {Z} (G_{i}/H_{i1}),} G i / H i 1 . {\displaystyle G_{i}/H_{i1}.} Niech H i 1 ⊴ H i 2 {\displaystyle H_{i1}\trianglelefteq H_{i2}} G i {\displaystyle G_{i}} φ : H 12 / H 11 → H 22 / H 21 {\displaystyle \varphi \colon H_{12}/H_{11}\to H_{22}/H_{21}} Wówczas H = { ( a , b ) ∈ H 12 × H 22 : φ ( a H 11 ) = b H 21 } {\displaystyle H={\big \{}(a,b)\in H_{12}\times H_{22}\colon \varphi (aH_{11})=bH_{21}{\big \}}} G 1 × G 2 . {\displaystyle G_{1}\times G_{2}.} Zakładając ponadto H i 1 , H i 2 ⊴ G {\displaystyle H_{i1},H_{i2}\trianglelefteq G} H i 2 / H i 1 ⊆ Z ( G i / H i 1 ) {\displaystyle H_{i2}/H_{i1}\subseteq \mathrm {Z} (G_{i}/H_{i1})} H ⊴ G 1 × G 2 . {\displaystyle H\trianglelefteq G_{1}\times G_{2}.} Konstrukcje podane w 1. i 2. są wzajemnie odwrotne. W literaturze[2]
Niech H {\displaystyle H} podgrupą w G 1 × G 2 {\displaystyle G_{1}\times G_{2}} π i : H → G i {\displaystyle \pi _{i}\colon H\to G_{i}} jądrach N i , {\displaystyle N_{i},} N i {\displaystyle N_{i}} podgrupą normalną w G i . {\displaystyle G_{i}.} obraz H {\displaystyle H} G 1 / N 1 × G 2 / N 2 {\displaystyle G_{1}/N_{1}\times G_{2}/N_{2}} wykresem izomorfizmu G 1 / N 1 ≃ G 2 / N 2 . {\displaystyle G_{1}/N_{1}\simeq G_{2}/N_{2}.} Diagram Hassego do lematu Zassenhausa (niestandardowo większe podgrupy znajdują się na dole diagramu, a mniejsze – na górze) obrazujący dodatkowo alternatywne określenie „motyli”.Niech G {\displaystyle G} B ⊴ A {\displaystyle B\trianglelefteq A} D ⊴ C {\displaystyle D\trianglelefteq C} B ( A ∩ D ) ⊴ B ( A ∩ C ) , {\displaystyle B(A\cap D)\trianglelefteq B(A\cap C),} D ( B ∩ C ) ⊴ D ( A ∩ C ) , {\displaystyle D(B\cap C)\trianglelefteq D(A\cap C),} B ( A ∩ C ) / B ( A ∩ D ) {\displaystyle B(A\cap C)/B(A\cap D)} D ( A ∩ C ) / D ( B ∩ C ) {\displaystyle D(A\cap C)/D(B\cap C)}
Dowód Zbiór H = { ( b c , d c ) ∈ G × G : b ∈ B , d ∈ D , c ∈ A ∩ C } {\displaystyle H=\{(bc,dc)\in G\times G\colon b\in B,d\in D,c\in A\cap C\}} G × G {\displaystyle G\times G} [e] H 12 = B ( A ∩ C ) {\displaystyle H_{12}=B(A\cap C)} H 22 = D ( A ∩ C ) {\displaystyle H_{22}=D(A\cap C)} G {\displaystyle G} H 11 = { a c : a ∈ B , c ∈ A ∩ D } = B ( A ∩ D ) {\displaystyle H_{11}=\{ac\colon a\in B,c\in A\cap D\}=B(A\cap D)} H 21 = D ( B ∩ C ) . {\displaystyle H_{21}=D(B\cap C).} H i 1 ⊴ H i 2 , {\displaystyle H_{i1}\trianglelefteq H_{i2},} B ( A ∩ D ) {\displaystyle B(A\cap D)} B ( A ∩ C ) , {\displaystyle B(A\cap C),} D ( B ∩ C ) {\displaystyle D(B\cap C)} D ( A ∩ C ) {\displaystyle D(A\cap C)} H 12 / H 11 {\displaystyle H_{12}/H_{11}} H 22 / H 21 {\displaystyle H_{22}/H_{21}}
↑ Kryterium bycia podgrupą : zbiór G 1 × G 2 {\displaystyle G_{1}\times G_{2}} ( e 1 , e 2 ) ∈ G 1 × G 2 {\displaystyle (e_{1},e_{2})\in G_{1}\times G_{2}} H 1 × H 2 ; {\displaystyle H_{1}\times H_{2};} h 1 , h 1 ′ ∈ H 1 {\displaystyle h_{1},h_{1}'\in H_{1}} h 2 , h 2 ′ ∈ H 2 , {\displaystyle h_{2},h_{2}'\in H_{2},} ( h 1 , h 2 ) , ( h 1 ′ , h 2 ′ ) ∈ H 1 × H 2 {\displaystyle (h_{1},h_{2}),(h_{1}',h_{2}')\in H_{1}\times H_{2}} ( h 1 , h 2 ) ( h 1 ′ , h 2 ′ ) − 1 = ( h 1 , h 2 ) ( h 1 ′ − 1 , h 2 ′ − 1 ) = ( h 1 h 1 ′ − 1 , h 2 h 2 ′ − 1 ) ∈ H 1 × H 2 , {\displaystyle (h_{1},h_{2})(h_{1}',h_{2}')^{-1}=(h_{1},h_{2})(h_{1}'^{-1},h_{2}'^{-1})=(h_{1}h_{1}'^{-1},h_{2}h_{2}'^{-1})\in H_{1}\times H_{2},} h 1 h 1 ′ − 1 ∈ H 1 {\displaystyle h_{1}h_{1}'^{-1}\in H_{1}} h 2 h 2 ′ − 1 ∈ H 2 . {\displaystyle h_{2}h_{2}'^{-1}\in H_{2}.} ↑ Nie każda podgrupa (normalna) iloczynu prostego dwóch grup jest podiloczynem: standardowym kontrprzykładem jest grupa czwórkowa Kleina Z 2 × Z 2 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}} { ( 0 ¯ , 0 ¯ ) , ( 1 ¯ , 1 ¯ ) } , {\displaystyle {\big \{}({\overline {0}},{\overline {0}}),({\overline {1}},{\overline {1}}){\big \}},} ( Z 2 , + ) {\displaystyle \mathbb {(} Z_{2},+)} Δ G = { ( g , … , g ) : g ∈ G } {\displaystyle \Delta _{G}=\{(g,\dots ,g)\colon g\in G\}} n {\displaystyle n} G n {\displaystyle G^{n}} ↑ Konieczność g i ∈ G i ∖ { e i } . {\displaystyle g_{i}\in G_{i}\setminus \{e_{i}\}.} ⟨ ( g 1 , g 2 ) ⟩ {\displaystyle {\big \langle }(g_{1},g_{2}){\big \rangle }} G 1 × G 2 , {\displaystyle G_{1}\times G_{2},} ⟨ ( g 1 , g 2 ) ⟩ = ⟨ g 1 ⟩ × ⟨ g 2 ⟩ . {\displaystyle {\big \langle }(g_{1},g_{2}){\big \rangle }=\langle g_{1}\rangle \times \langle g_{2}\rangle .} g i {\displaystyle g_{i}} Dostateczność H {\displaystyle H} G 1 × G 2 {\displaystyle G_{1}\times G_{2}} ( g 1 , g 2 ) ∈ H . {\displaystyle (g_{1},g_{2})\in H.} g 1 , g 2 {\displaystyle g_{1},g_{2}} ⟨ ( g 1 , g 2 ) ⟩ = ⟨ g 1 ⟩ × ⟨ g 2 ⟩ , {\displaystyle {\big \langle }(g_{1},g_{2}){\big \rangle }=\langle g_{1}\rangle \times \langle g_{2}\rangle ,} ( g 1 , e 2 ) , ( e 1 , g 2 ) ∈ H , {\displaystyle (g_{1},e_{2}),(e_{1},g_{2})\in H,} H = H 1 × H 2 , {\displaystyle H=H_{1}\times H_{2},} H 1 = { g ∈ G 1 : ( g , e 2 ) ∈ H } {\displaystyle H_{1}={\big \{}g\in G_{1}\colon (g,e_{2})\in H{\big \}}} H 2 = { g ∈ G 2 : ( e 1 , g ) ∈ H } . {\displaystyle H_{2}={\big \{}g\in G_{2}\colon (e_{1},g)\in H{\big \}}.} ↑ Konieczność G 1 × G 2 {\displaystyle G_{1}\times G_{2}} G 1 × G 2 = ⟨ ( g 1 , g 2 ) ⟩ . {\displaystyle G_{1}\times G_{2}={\big \langle }(g_{1},g_{2}){\big \rangle }.} g ∈ G 1 {\displaystyle g\in G_{1}} ( g , e 2 ) = ( g 1 , g 2 ) n {\displaystyle (g,e_{2})=(g_{1},g_{2})^{n}} n . {\displaystyle n.} g = g 1 n {\displaystyle g=g_{1}^{n}} g 2 n = e 2 ; {\displaystyle g_{2}^{n}=e_{2};} G 1 = ⟨ g 1 ⟩ , {\displaystyle G_{1}=\langle g_{1}\rangle ,} g 2 {\displaystyle g_{2}} g 1 {\displaystyle g_{1}} G 2 = ⟨ g 2 ⟩ . {\displaystyle G_{2}=\langle g_{2}\rangle .} o r d ( g i ) = n i {\displaystyle \mathrm {ord} (g_{i})=n_{i}} g i . {\displaystyle g_{i}.} o r d ( ( g 1 , g 2 ) ) = | G 1 × G 2 | = n 1 n 2 . {\displaystyle \mathrm {ord} {\big (}(g_{1},g_{2}){\big )}=|G_{1}\times G_{2}|=n_{1}n_{2}.} l = n w w ( n 1 , n 2 ) , {\displaystyle l=\mathrm {nww} (n_{1},n_{2}),} ( g 1 , g 2 ) l = ( e 1 , e 2 ) ; {\displaystyle (g_{1},g_{2})^{l}=(e_{1},e_{2});} l = n 1 n 2 , {\displaystyle l=n_{1}n_{2},} n w d ( n 1 , n 2 ) = 1. {\displaystyle \mathrm {nwd} (n_{1},n_{2})=1.} Dostateczność G i = ⟨ g i ⟩ , {\displaystyle G_{i}=\langle g_{i}\rangle ,} | G i | = n i , {\displaystyle |G_{i}|=n_{i},} n w d ( n 1 , n 2 ) = 1. {\displaystyle \mathrm {nwd} (n_{1},n_{2})=1.} ( g 1 , g 2 ) n 1 n 2 = ( ( g 1 n 1 ) n 2 , ( g 2 n 2 ) n 1 ) = ( e 1 , e 2 ) . {\displaystyle (g_{1},g_{2})^{n_{1}n_{2}}=\left((g_{1}^{n_{1}})^{n_{2}},(g_{2}^{n_{2}})^{n_{1}}\right)=(e_{1},e_{2}).} ( e 1 , e 2 ) = ( g 1 , g 2 ) l {\displaystyle (e_{1},e_{2})=(g_{1},g_{2})^{l}} g i l = e i ; {\displaystyle g_{i}^{l}=e_{i};} n i | l . {\displaystyle n_{i}|l.} n w d ( n 1 , n 2 ) = 1 , {\displaystyle \mathrm {nwd} (n_{1},n_{2})=1,} n 1 n 2 | l . {\displaystyle n_{1}n_{2}|l.} ( g 1 , g 2 ) {\displaystyle (g_{1},g_{2})} n 1 n 2 , {\displaystyle n_{1}n_{2},} ⟨ ( g 1 , g 2 ) ⟩ = G 1 × G 2 . {\displaystyle {\big \langle }(g_{1},g_{2}){\big \rangle }=G_{1}\times G_{2}.} ↑ Niech ( b c , d c ) , ( b ′ c ′ , d ′ c ′ ) ∈ H , {\displaystyle (bc,dc),(b'c',d'c')\in H,} b , b ′ ∈ B , {\displaystyle b,b'\in B,} d , d ′ ∈ D {\displaystyle d,d'\in D} c , c ′ ∈ A ∩ C . {\displaystyle c,c'\in A\cap C.} B ⊴ A {\displaystyle B\trianglelefteq A} c b ′ = b ¯ c {\displaystyle cb'={\overline {b}}c} c − 1 b − 1 = b ∗ c − 1 {\displaystyle c^{-1}b^{-1}=b^{*}c^{-1}} b ¯ , b ∗ ∈ B {\displaystyle {\overline {b}},b^{*}\in B} D ⊴ C {\displaystyle D\trianglelefteq C} c d ′ = d ¯ c {\displaystyle cd'={\overline {d}}c} c − 1 d − 1 = d ∗ c − 1 {\displaystyle c^{-1}d^{-1}=d^{*}c^{-1}} d ¯ , d ∗ ∈ D . {\displaystyle {\overline {d}},d^{*}\in D.} ( b c , d c ) ( b ′ c ′ , d ′ c ′ ) = ( b c b ′ c ′ , d c d ′ c ′ ) = ( b b ¯ c c ′ , d d ¯ c c ′ ) ∈ H {\displaystyle (bc,dc)(b'c',d'c')=(bcb'c',dcd'c')=(b{\overline {b}}cc',d{\overline {d}}cc')\in H} ( b c , d c ) − 1 = ( c − 1 b − 1 , c − 1 d − 1 ) = ( b ∗ c − 1 , d ∗ c − 1 ) ∈ H . {\displaystyle (bc,dc)^{-1}=(c^{-1}b^{-1},c^{-1}d^{-1})=(b^{*}c^{-1},d^{*}c^{-1})\in H.} ↑ Édouard Goursat . Sur les substitutions orthogonales et les divisions régulières de l’espace ISSN 0012-9593 (fr. ) . ↑ Serge S. Lang Serge S. Algebra , wyd. 3, t. 211, Nowy Jork: Springer-Verlag, 2002 (Graduate Texts in Mathematics ), DOI : 10.1007/978-1-4613-0041-0 , ISBN 978-0-387-95385-4 , ISSN 0072-5285 .brak strony (książka) 
