А.В. Михалёв - Лекции по высшей алгебре (1106006), страница 2

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DZãáâì H — ¯®¤£à㯯 £à㯯ë G, à áᬮâਬ ¯®«¨£®­ MH = GH , (xh) → xh¤«ï x ∈ G, h ∈ H. …᫨ p, x ∈ G, â® St(p) = {h ∈ H|ph = p} = {e} ¨ Orb(x) = xH — ¯à ¢ë©á¬¥­ë© ª« áá xH ¯® ¯®¤£à㯯¥ H, ¯®à®¤ñ­­ë© í«¥¬¥­â®¬ x ∈ G. ’ ª¨¬ ®¡à §®¬, ¢ í⮬ç áâ­®¬ á«ãç ¥ ¯®«¨£®­ GH à §¡¨¥­¨¥ ­ ®à¡¨âë ¯à¥¢à é ¥âáï ¢ å®à®è® §­ ª®¬®¥ ­ ¬. à §¡¨¥­¨¥£àã¯¯ë ¢ ®¡ê¥¤¨­¥­¨¥ ­¥¯¥à¥á¥ª îé¨åáï à §«¨ç­ëå ¯à ¢ëå ᬥ­ëå ª« áᮢ G = ∪ xH, ¨ ª ªá«¥¤á⢨¥ ¯®¤áçñâ í«¥¬¥­â®¢ ¯® ®à¡¨â ¬ ¨¬¥¥¬|G| =(â. ¥.X| Orb(x)| =X|xH| = [G : H]|H|⥮६㠋 £à ­ ). §¡¨¥­¨¥ ­ «¥¢ë¥ ᬥ­ë¥ ª« ááë Hx ¯® ¯®¤£à㯯¥ H £à㯯ë G ¯®«ãç ¥âáï à áᬮâ७¨¥¬«¥¢®£® ¯®«¨£®­ H G.DZਬ¥à 8 (¤¥©á⢨¥ £à㯯ë G ¯à ¢ë¬¨ 㬭®¥­¨ï¬¨ ­ ¬­®¥á⢥ «¥¢ëå ᬥ­ë媫 áᮢ).

DZãáâì H — ¯®¤£à㯯 £à㯯ë G,MG = {Hx, x ∈ G},(Hx)g = Hxg¤«ï x, g ∈ G(í⮠㬭®¥­¨¥ ª®à४⭮: ¥á«¨ Hx = Hx′ , â® x′ = hx, h ∈ H ¨ ¯®í⮬ã x′ g = hxg, ⮣¤ Hx′ g =Hxg). …᫨ x, g1 , g2 ∈ G, â® (Hx)(g1 , g2 ) = ((Hx)g1 )g2 ¨ (Hx)e = Hx, â. ¥. MG = {Hx|x ∈ G}G —¯à ¢ë© ¯®«¨£®­ ­ ¤ £à㯯®© G.

’ ª ª ª Orb(Hx) = MG , â® íâ® ¤¥©á⢨¥ £à㯯ë G âà ­§¨â¨¢­®.9. DZãáâì k ∈ N ¨ MG = Lk (G)G = {X ⊆ G||X| = k}, (X, g) → Xg ¤«ï X ∈ Lk (G),g ∈ G (ïá­®, çâ® |Xg| = |X| = k). Ÿá­®, çâ® MG = Lk (G)G — ¯à ¢ë© G-¯®«¨£®­.‚ ç áâ­®áâ¨, ¥á«¨ |G| = n < ∞ ¨ k ∈ N, â® MG = Lk (G)G = {X1 , . . . , XCnk ||Xi| = k}G — ¯à ¢ë©G-¯®«¨£®­.10 (GG). DZãáâì MG = GG , (m, g) →g −1 mg = mαg ¤«ï m, g ∈ G. ’ ª ª ª ¤«ï m, g, h ∈ G ¨¬¥¥¬DZਬ¥àDZਬ¥à¤¥©á⢨¥ £àã¯¯ë ­ £à㯯¥ ᮯà省¨¥¬mαgh = (gh)−1 m(gh) = h−1 (g −1 mg)h = (mαg )αh ,â. ¥.

αgh = αg αh ,mαe = e−1 me = m,â® MG = GG á ᮯà省¨¥¬ — ¯à ¢ë© G-¯®«¨£®­.‡ ¬¥ç ­¨¥. Š ª ¬ë ®â¬¥ç «¨, αg : G → G, m → mαg = g−1mg, — ¢â®¬®à䨧¬ £à㯯ë G,­ §ë¢ ¥¬ë© ¢­ãâ७­¨¬ ¢â®¬®à䨧¬®¬ £à㯯ë G. ‚ᥠ¢­ãâ७­¨¥ ¢â®¬®à䨧¬ë In Aut(G) =Inr Aut(G) = {αg |g ∈ G} — ¯®¤£à㯯 £à㯯ë Aut(G) ¢â®¬®à䨧¬®¢ £à㯯ë G, ª®â®à ï ¢ ᢮î®ç¥à¥¤ì ï¥âáï ¯®¤£à㯯®© S r (G) = S(G) ¢á¥¬ ¯®¤áâ ­®¢®ª ­ ¬­®¥á⢥ G.’ ª¨¬ ®¡à §®¬, ¬ë ¨¬¥¥¬ á«¥¤ãî騥 ¯®«¨£®­ë:GS(G) ,GAut G ,GIn Aut G .Š®¬¯®§¨æ¨ï £®¬®¬®à䨧¬ G → In Aut G, g → αg , á ¥áâ¥á⢥­­ë¬ ¯à¥¤áâ ¢«¥­¨¥¬ ¯à ¢ë¬ ¤¥©á⢨¥¬ £à㯯ë In Aut G ­ ¬­®¥á⢥ G ᮢ¯ ¤ ¥â á ¤¥©á⢨¥¬ £à㯯ë G ᮯà省¨ï¬¨ ­ £à㯯¥G.5‹¥ªæ¨ï 8http://mmresource.nm.ru/“¯à ­¥­¨¥. Žâ®¡à ¥­¨¥ α : S3 → Aut S3, g → αg , ï¥âáï ¨§®¬®à䨧¬®¬ £à㯯.Š« áá ᮯà省­ëå í«¥¬¥­â®¢.…᫨ x ∈ G, â® ¯à¨ ᮯà省¨¨Orb(x) = {g −1 xg|g ∈ G} —ª« ááë ᮯàïñ­­ëå í«¥¬¥­â®¢ í«¥¬¥­â x.Ÿá­®, çâ® Orb(e) = {e}. ®«¥¥ ⮣®, | Orb(x)| = 1 ⇔ x ∈ Z(G), â.

¥. ®¤­®í«¥¬¥­â­ë¥ ®à¡¨âë —íâ®, ¢ â®ç­®áâ¨, í«¥¬¥­âë 業âà , ¯®áª®«ìªã g−1 xg = x ¤«ï ¢á¥å g ∈ G à ¢­®á¨«ì­® ⮬ã, çâ®xg = gx ¤«ï ¢á¥å g ∈ G, â. ¥. ⮬ã, çâ® x ∈ Z(G).Ÿá­®, çâ®St(x) = {g|g −1xg = x} = C(x),£¤¥ C(x) = {y ∈ G|xg = gx) — 業âà «¨§ â®à í«¥¬¥­â x ∈ G.’ ª¨¬ ®¡à §®¬, ⥮६ ® à §¡¨¥­¨¨ ­ ®à¡¨âë ¢ ¤ ­­®¬ á«ãç ¥ ®§­ ç ¥â á«¥¤ãî饥:’¥®à¥¬ (® à §¡¨¥­¨¨ ­ ª« ááë ᮯàïñ­­ëå í«¥¬¥­â®¢).DZãáâì G — £à㯯 , ⮣¤ :£à㯯 ï¥âáï ®¡ê¥¤¨­¥­¨¥¬ ®à¡¨â — ­¥¯¥à¥á¥ª îé¨åáï à §«¨ç­ëå ª« áᮢ ᮯàïñ­­ëå í«¥¬¥­â®¢ (â.

¥. ®â­®è¥­¨¥ ᮯàïñ­­®á⨠y ∼ x, ¥á«¨, y = g−1 xg, ï¥âáï ®â­®è¥­¨¥¬íª¢¨¢ «¥­â­®áâ¨);2) ç¨á«® í«¥¬¥­â®¢ ª®­¥ç­®© £à㯯ë G, ᮯàïñ­­ëå á í«¥¬¥­â®¬ x ∈ G, à ¢­® ¨­¤¥ªáã 業âà «¨§ â®à C(x) í«¥¬¥­â x ∈ G ¢ £à㯯¥ (¯®áª®«ìªã |G| = | Orb(x)| · | St(x)| = { ç¨á«® ᮯàïñ­­ëåá x í«¥¬¥­â®¢ } · |C(x)|), â. ¥. ç¨á«ã |G|/|C(x)|.1)‡ ¤ ç .  §¡¨¥­¨¥ ­ ª« ááë ᮯàïñ­­ëå í«¥¬¥­â®¢ ¢ £à㯯¥ ¯®¤áâ ­®¢®ª Sn ®¯à¥¤¥«ï¥âáï⨯®¬ 横«®¢®£® à §«®¥­¨ï (â. ¥. ¤¢¥ ¯®¤áâ ­®¢ª¨ ¢ Sn ᮯà省ë ⮣¤ ¨ ⮫쪮 ⮣¤ , ª®£¤ ®­¨ ¨¬¥îâ ®¤¨­ ª®¢ë¥ 横«®¢ë¥ à §«®¥­¨ï, â. ¥.

¤«ï ª ¤®£® ç¨á« r ®¤¨­ ª®¢®¥ ç¨á«® 横«®¢¤«¨­ë r ¢ ¨å 横«®¢ëå à §«®¥­¨ïå).’¥®à¥¬ .–¥­âà ª®­¥ç­®© p-£à㯯ë.DZãáâì G — ª®­¥ç­ ï p-£à㯯 , â. ¥. |G| = pk , £¤¥ p — ¯à®á⮥ ç¨á«®, k ∈ N. ’®£¤ ¥ñ 業âà ­¥âਢ¨ «¥­, â. ¥. |Z(G)| > 1.„®ª § ⥫ìá⢮.

 áᬮâਬ à §¡¨¥­¨¥ £à㯯ë G ­ ª« ááë ᮯàïñ­­ëå í«¥¬¥­â®¢. Ž¤­®-í«¥¬¥­â­ë© ª« áá — íâ® ¢ â®ç­®áâ¨ í«¥¬¥­â 業âà (®¤¨­ ¨§ ­¨å {e}). ‘®¤¥à 騩 ¡®«ìè¥ ®¤­®£®í«¥¬¥­â ª« áá ᮯàïñ­­ëå í«¥¬¥­â®¢ ᮤ¥à â pl í«¥¬¥­â®¢, £¤¥ l > 1 (ª ª ­¥âਢ¨ «ì­ë© ¤¥«¨â¥«ì ç¨á« |G| = pk ).Žâáî¤ á«¥¤ã¥â, çâ® |Z(G)| > 1 (¢ ¯à®â¨¢­®¬ á«ãç ¥, pk = 1 + pq). ’¥®à¥¬ (® ª®¬¬ãâ ⨢­®á⨠£àã¯¯ë ¨§ p2 í«¥¬¥­â®¢).DZãáâì G — ª®­¥ç­ ï £à㯯 ,£¤¥ p — ¯à®á⮥ ç¨á«®. ’®£¤ G — ¡¥«¥¢ (â.

¥. ª®¬¬ãâ ⨢­ ï) £à㯯 .„®ª § ⥫ìá⢮. ‚ ᨫ㠯।ë¤ã饩 ⥮६ë, |Z(G)| > 1, â. ¥. |Z(G)| = p ¨«¨ |Z(G)| = p2 . ®|G| = p2 ,¯¥à¢ë© á«ãç © (|Z(G)| = p) ­¥¢®§¬®¥­, ¯®áª®«ìªã ¢ í⮬ á«ãç ¥ |Z/Z(G)| = p2 /p = p, ¨ ¯®í⮬ãZ/Z(G) — 横«¨ç¥áª ï £à㯯 , çâ®, ª ª ¡ë«® ¯®ª § ­® à ­¥¥, ­¥¢®§¬®­®. ˆâ ª, |Z(G)| = p2 ,â. ¥. G = Z(G), ¨ ¯®í⮬㠣à㯯 G ª®¬¬ãâ ⨢­ .6‹¥ªæ¨ï 9http://mmresource.nm.ru/‹…Š–ˆŸ9.28®ªâï¡àï 2002 £.’¥®à¥¬ë ‘¨«®¢ .Ž¤­¨¬ ¨§ ïનå १ã«ìâ ⮢ ⥮ਨ ª®­¥ç­ëå £à㯯 ¢ ­ ¯à ¢«¥­¨¨ ç áâ¨ç­®£® ®¡à 饭¨ï â¥®à¥¬ë ‹ £à ­ ïîâáï á«¥¤ãî騥 â¥®à¥¬ë ‘¨«®¢ (1872).’¥®à¥¬ (1-ï ⥮६ ‘¨«®¢ ® áãé¥á⢮¢ ­¨¨ ᨫ®¢áª¨å ¯®¤£à㯯).

DZãáâì G — ª®­¥ç­ ï £à㯯 , |G| = n = pk m, k ≥ 1 ,p — ¯à®á⮥ ç¨á«®, (p, m) = 1. ’®£¤ £à㯯 G ᮤ¥à¨â¯®¤£à㯯ã H â ªãî, çâ® |H| = pk (â ª ï ¯®¤£à㯯 ­ §ë¢ ¥âáï ᨫ®¢áª®© ¯®¤£à㯯®© £à㯯ë G).„®ª § ⥫ìá⢮.…᫨ G — ¡¥«¥¢ £à㯯 , |G| = pk m, (p, m) = 1, â® ¢ ª ç¥á⢥ H ¬®­® ¢§ïâì ¯à¨¬ à­ã¬¯®­¥­âã Gp £à㯯ë G (â.

¥. ¯àï¬ãî á㬬㠢á¥å p-¯à¨¬ à­ëå 横«¨ç¥áª¨å £à㯯 ª ­®­¨ç¥áª®£®à §«®¥­¨ï), ¨ ⮣¤ Gp = pk .2) …᫨ |G| = pk (â. ¥. m = 1), â® G = H.3) DZ஢¥¤¥¬ ¤®ª § ⥫ìá⢮ ¨­¤ãªâ¨¢­®.‘«ãç © 1. p ¤¥«¨â ç¨á«® |Z(G)| í«¥¬¥­â®¢ 業âà Z(G) £à㯯ë G. ˆ§ ®¡à 饭¨ï â¥®à¥¬ë‹ £à ­ ¤«ï ¡¥«¥¢ëå £à㯯 ­ ©¤¥âáï ¯®¤£à㯯 A ¢ 業âॠZ(G), |A| = p. Ÿá­®, çâ® A ⊳ G,|G/A| = n/p = pk−1 m < n. ‚ ᨫ㠨­¤ãªâ¨¢­®£® ¯à¥¤¯®«®¥­¨ï, ¢ Ḡ = G/A ­ ©¤¥âáï ¯®¤£à㯯 B̄, |B̄| = pk−1 .

® B̄ = B/A ⊂ G/A, £¤¥ A ⊆ B ⊆ G, ¯®í⮬ã |B| = |A||B/A| = ppk−1 = pk , â. ¥. B —ᨫ®¢áª ï ¯®¤£à㯯 £à㯯ë G.‘«ãç © 2. p ­¥ ¤¥«¨â ¯®à冷ª |Z(G)| 業âà Z(G) £à㯯ë G.  áᬮâਬ à §«®¥­¨¥ £à㯯ë·∪ Ci . DZãáâì C1 , . . . , Cr — ®¤­®í«¥¬¥­â­ë¥ ª« ááë­ ª« ááë ᮯà省­ëå í«¥¬¥­â®¢ G = i=1,...,lᮯà省­ëå í«¥¬¥­â®¢ (â. ¥. ¢á¥ í«¥¬¥­âë 業âà Z(G), r = |Z(G)|). ’ ª ª ª |G| ¤¥«¨âáï ­ p, ç¨á«® r ­¥ ¤¥«¨âáï ­ p, ­ ©¤¥âáï ®à¡¨â Ci = Orb(xi ), r + 1 ≤ i ≤ l, â ª ï, çâ® |G|/|C(xi )| = |Ci | ­¥¤¥«¨âáï ­ p.

’®£¤ |C(xi )| < n, â® ¯® ¨­¤ãªâ¨¢­®¬ã ¯à¥¤¯®«®¥­¨î ¢ C(xi ) ­ ©¤¥âáï ¯®¤£à㯯 H, â ª ï, çâ® |H| = pk , â. ¥. H — ᨫ®¢áª ï ¯®¤£à㯯 £à㯯ë G (H ⊆ C(xi ) ⊆ G). 1)’¥®à¥¬ (2-ï ⥮६ ‘¨«®¢ ® ᮯà省­®á⨠ᨫ®¢áª¨å ¯®¤£à㯯). DZãáâì G —ª®­¥ç­ ï £à㯯 , |G| = pk m, k ≥ 1, (p, m) = 1. ) ‹î¡ ï p−¯®¤£à㯯 H £à㯯ë G (â. ¥. |H| = pl , l ≤ k) ᮤ¥à¨âáï ¢ ­¥ª®â®à®© ᨫ®¢áª®©p−¯®¤£à㯯¥.¡) ‹î¡ë¥ ¤¢¥ ᨫ®¢áª¨¥ ¯®¤£à㯯ë S1 ¨ S2 ᮯà省ë (â. ¥.

S2 = g−1 S1 g ¤«ï ­¥ª®â®à®£® g ∈ G).„®ª § ⥫ìá⢮. ‘«ãç ©, ª®£¤ m = 1 ïᥭ. DZãáâì m > 1 ¨ ¯ãáâì S, |S| = pk , — ᨫ®¢áª ïp−¯®¤£à㯯 (áãé¥á⢮¢ ­¨¥ ª®â®à®© ¤®ª § ­® ¢ 1-®© ⥮६¥ ‘¨«®¢ ).  áᬮâਬ á«¥¤ãî騩¯à ¢ë© H−¯®«¨£®­: MH = {Sx|x ∈ G}, (Sx, a) → Sxa ¤«ï x ∈ G, a ∈ H (â. ¥. «¥¢ë¥ ᬥ­ë¥ ª« ááë Sx ¯®¤£à㯯ë S c 㬭®¥­¨¥¬ á¯à ¢ ­ í«¥¬¥­âë ¨§ ¯®¤£à㯯ë H); ª®à४⭮áâì 㬭®¥­¨ïïá­ : Sx = Sx′ ⇒ x′ = sx, ⇒ x′ a = s(xa) ⇒ Sx′ a = Sxa.

ˆ§ â¥®à¥¬ë ‹ £à ­ ¤«ï ¯®¤£à㯯ëS: |M | = |G|/|S| = pk m/pk = m > 1, ¯à¨ í⮬ (p, m) = 1. ’ ª ª ª pl = |H| = | St(y)|| Orb(y)| ¤«ïí«¥¬¥­â y ∈ MH , â® ç¨á«® í«¥¬¥­â®¢ ¢ ª ¤®© ­¥®¤­®í«¥¬¥­â­®© ®à¡¨â¥ ¯®«¨£®­ MH ¤¥«¨âáï­ p. ‘«¥¤®¢ ⥫쭮, áãé¥áâ¢ã¥â ®¤­®í«¥¬¥­â­ ï ®à¡¨â Sx ∈ MH , x ∈ G, â. ¥. ¤«ï Sx ¨¬¥¥¬SxH = Sx. ® ⮣¤ xH ⊆ Sx, ¨ ¯®í⮬ã H ⊆ x−1 Sx. ’ ª ª ª |x−1 Sx| = |S| = pk , â® x−1 Sxï¥âáï ᨫ®¢áª®© p−¯®¤£à㯯®©, ᮤ¥à 饩 ¨á室­ãî p−¯®¤£à㯯ã H.…᫨ ¥ H — ᨫ®¢áª ï p−¯®¤£à㯯 , â.

¥. |H| = pk , â® H = x−1 Sx. ’¥¬ á ¬ë¬ ¯®ª § ­®, ç⮫î¡ë¥ ¤¢¥ ᨫ®¢áª¨¥ ¯®¤£à㯯ë S1 = S ¨ S2 = H ᮯàï¥­ë ¬¥¤ã ᮡ®©. ®à¬ «¨§ â®à ¯®¤£à㯯ë.DZãáâì LG = L(G) — ᮢ®ªã¯­®áâì ¢á¥å ¯®¤£à㯯 H £à㯯ë G, (H, g) → g−1Hg, g → G.Ž¯à¥¤¥«¥­¨¥. ‘â ¡¨«¨§ â®à ¯®¤£à㯯ë H ¯à¨ í⮬ ¤¥©á⢨¨ £à㯯ë G ᮯà省¨ï¬¨ St(H) ={g ∈ G|g −1 Hg = H} ­ §ë¢ ¥âáï ­®à¬ «¨§ â®à®¬ ¯®¤£à㯯ë H ¢ £à㯯¥ G (¤«ï ­¥£® ¨á¯®«ì§ã¥âáﮡ®§­ 祭¨¥ NG(H)).7‹¥ªæ¨ï 9http://mmresource.nm.ru/ˆâ ª, NG(H) = St(H) = {g → G|g−1 Hg = H}.‹¥¬¬ ‘¢®©á⢠­®à¬ «¨§ â®à NG(H).1) NG (H) — ¯®¤£à㯯 £à㯯ë G ᮤ¥à é ï ¯®¤£à㯯ã H;2) H ⊳ NG (H);3) ¥á«¨ H ⊳ K ⊆ G (â.

¥. K — ¯®¤£à㯯 £à㯯ë G, ᮤ¥à é ï ¯®¤£à㯯ã H, ¨ H —­®à¬ «ì­ ï ¯®¤£à㯯 ¢ K), â® K ⊆ NG(H), ¨, â ª¨¬ ®¡à §®¬, NG (H) — ­ ¨¡®«ìè ï ¯®¤£à㯯 ,ᮤ¥à é ï H ¢ ª ç¥á⢥ ­®à¬ «ì­®© ¯®¤£à㯯ë (ª®­¥ç­®, ¥á«¨ H ⊳ G, â® NG(H) = G).„®ª § ⥫ìá⢮.1) ’ ª ª ª NG (H) = St(H), â® ïá­®, çâ® NG (H) (ª ª «î¡®© áâ ¡¨«¨§ â®à) — ¯®¤£à㯯 £à㯯ëG.2) …᫨ g → NG (H), â® g −1 Hg = H, â. ¥. H ⊳ NG (H) (â. ¥. H — ­®à¬ «ì­ ï ¯®¤£à㯯 £à㯯ëNG (H)).3) …᫨ H ⊳ K ⊆ G, â® ¤«ï «î¡®£® í«¥¬¥­â g ∈ K ¨¬¥¥¬ g −1 Hg = H (¯®áª®«ìªã H —­®à¬ «ì­ ï ¯®¤£à㯯 £àã¯¯ë Š), â.

¥. g ∈ St(H) = NG (H), ¨ ¯®í⮬ã K ⊆ NG(H). (). DZãáâì G — ª®­¥ç­ ï£à㯯 , n = |G| = pk m, k ≥ 1, (p, m) = 1. —¥à¥§ n(p) ®¡®§­ 稬 ç¨á«® ᨫ®¢áª¨å p-¯®¤£à㯯. ’®£¤ :1) n(p) — ¤¥«¨â¥«ì ç¨á« n = |G|;2) n(p) = 1 + pq (â. ¥. ®áâ ⮪ ¯à¨ ¤¥«¥­¨¨ ç¨á« n(p) ­ ¯à®á⮥ ç¨á«® p à ¢­® 1).„®ª § ⥫ìá⢮. 1)  áᬮâਬ ¯à ¢ë© G-¯®«¨£®­’¥®à¥¬ ’à¥âìï ⥮६ ‘¨«®¢ ® ç¨á«¥ ᨫ®¢áª¨å ¯®¤£à㯯MG = L(G) = {H|H ⊆ G},(â.

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