❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ◆●❯❨➍◆ ❚❍➚ ❚❍❷❖ ◆❍➶▼ ❍Ú❯ ❍❸◆ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✼ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ◆●❯❨➍◆ ❚❍➚ ❚❍❷❖ ◆❍➶▼ ❍Ú❯ ❍❸◆ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ✣↕✐ sè ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚✐➳♥ s➽✿ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ❍➔ ◆ë✐ ✕ ◆➠♠ ✷✵✶✼ ▼ö❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✶ ▲í✐ ❝❛♠ ✤♦❛♥ ✷ ▲í✐ ♥â✐ ✤➛✉ ✸ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✺ ✶✳✶ ◆❤â♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ◆❤â♠ ❝♦♥ ✈➔ ✤✐➲✉ ❦✐➺♥ t÷ì♥❣ ✤÷ì♥❣ ✳ ✳ ✳ ✳ ✶✳✸ ◆❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ✈➔ ♥❤â♠ t❤÷ì♥❣ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ◆❤â♠ ❝♦♥ ❝❤✉➞♥ t➢❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❳➙② ❞ü♥❣ ♥❤â♠ t❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ỗ õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❚➟♣ s✐♥❤ ❝õ❛ ♥❤â♠✱ ♥❤â♠ ①②❝❧✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺✳✶ ✣à♥❤ ♥❣❤➽❛ t➟♣ s✐♥❤ ❝õ❛ ♥❤â♠ ✳ ✳ ✳ ✳ ✶✳✺✳✷ ✣à♥❤ ♥❣❤➽❛ ♥❤â♠ ①②❝❧✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✻ ❈➜♣ ❝õ❛ ♥❤â♠✱ ❝➜♣ ❝õ❛ ♣❤➛♥ tû tr♦♥❣ ♥❤â♠ ✶✳✼ ❚➼❝❤ trü❝ t✐➳♣✱ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♥❤â♠ ✶✳✼✳✶ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✻ ✼ ✼ ✽ ✽ ✶✵ ✶✵ ✶✶ ✶✶ ✶✷ ✶✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ✷ ◆❤â♠ ❤ú✉ ❤↕♥ ✷✳✶ ◆❤â♠ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✣à♥❤ ❧➼ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ✣à♥❤ ❧➼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❚ê♥❣ q✉→t ❤â❛ ❝õ❛ ✤à♥❤ ❧➼ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✸ ❍➺ q✉↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ◆❤â♠ ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ◆❤â♠ t❤❛② ♣❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶ ✣à♥❤ ♥❣❤➽❛ ♥❤â♠ t❤❛② ♣❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✷ ❚➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ◆❤â♠ ◗✉❛t❡r♥✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳✶ ❇ê ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳✷ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ ◆❤â♠ ❆❜❡❧ ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✷ P❤➛♥ tû t✉➛♥ ❤♦➔♥✱ ♥❤â♠ ❝♦♥ ①♦➢♥✱ ♥❤â♠ t✉➛♥ ❤♦➔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ▼ët sè ❜➔✐ t➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❇✐➸✉ ❞✐➵♥ ♥❤â♠ ❤ú✉ ❤↕♥ ✸✳✶ ✸✳✷ ✸✳✸ ✸✳✹ ❇✐➸✉ ❞✐➵♥ ♥❤â♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇ê ✤➲ ❙❝❤✉r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✹ ✶✹ ✶✹ ✶✺ ✶✻ ✶✽ ✶✽ ✶✾ ✷✸ ✷✹ ✷✺ ✷✻ ✷✻ ✷✾ ✷✾ ✷✾ ✷✾ ✸✺ ✹✸ ✹✸ ✹✼ ✺✶ ✺✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ✸✳✺ ❙è ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ❑➳t ❧✉➟♥ ✻✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✻✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ▲í✐ ❝↔♠ ì♥ ❙❛✉ ♠ët t❤í✐ ❣✐❛♥ ❞➔✐ ♥❣❤✐➯♠ tó❝✱ ♠✐➺t ♠➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝ị♥❣ ✈ỵ✐ sü ❣✐ó♣ ✤ï t➟♥ t➻♥❤ ❝õ❛ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥✳ ✣➳♥ ♥❛②✱ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ✤➣ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤✱ s➙✉ s➢❝ tỵ✐ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tê ✣↕✐ sè✱ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✤➦❝ ❜✐➺t ❧➔ ❝æ ❣✐→♦ ✲ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ♥❣÷í✐ ✤➣ trü❝ t✐➳♣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï✱ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝❤♦ ❡♠ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉✱ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❉♦ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ❝ơ♥❣ ♥❤÷ ❦✐➳♥ t❤ù❝ ❝õ❛ ❜↔♥ t❤➙♥ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❑➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü õ ỵ t ổ s ✈✐➯♥✳ ▼ët ❧➛♥ ♥ú❛ ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✼ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ▲í✐ ❝❛♠ ✤♦❛♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✧◆❤â♠ ❤ú✉ ❤↕♥✧ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❞♦ sü ❝è ❣➢♥❣✱ ♥é ❧ü❝ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝ị♥❣ ✈ỵ✐ sü ❣✐ó♣ ✤ï t➟♥ t➻♥❤ ❝õ❛ ❝ỉ ❣✐→♦ ✲ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ✳ ❚r♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ♥❤÷ ✤➣ ✈✐➳t tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❱➻ ✈➟②✱ ❡♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❦➳t q✉↔ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ✈ỵ✐ ❦➳t q✉↔ ❝õ❛ t→❝ ❣✐↔ ♥➔♦ ❦❤→❝✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✼ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ▲í✐ ♥â✐ ✤➛✉ ✣↕✐ sè ❧➔ ♠ët ❜ë ♣❤➟♥ q✉❛♥ trå♥❣ ❝õ❛ ❚♦→♥ ❤å❝✳ ❚r♦♥❣ ✤â✱ ♥❤â♠ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✤è✐ t÷đ♥❣ ❝ì ❜↔♥ ♥❤➜t ✈➔ ❝ê ✤✐➸♥ ♥❤➜t ❝õ❛ ✣↕✐ sè✳ ✧◆❤â♠ ❤ú✉ ❤↕♥✧ ❧➔ ♥ë✐ ❞✉♥❣ q✉❛♥ trå♥❣ ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ❝→❝ ♥❣➔♥❤ t♦→♥ ❤å❝ ❝ơ♥❣ ♥❤÷ tr♦♥❣ t❤ü❝ t✐➵♥✳ ❱ỵ✐ ❧á♥❣ ②➯✉ t❤➼❝❤✱ ♥✐➲♠ s❛② ♠➯ ♠✉è♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ t➻♠ ❤✐➸✉ ✈➲ ✣↕✐ sè ♥â✐ ❝❤✉♥❣ ✈➔ ❝➜✉ tró❝ ♥❤â♠ ♥â✐ r✐➯♥❣✱ ❡♠ ✤➣ ❝❤å♥ tõ ỳ õ ỗ ✸ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♥❤â♠✳ ❈❤÷ì♥❣ ✷✿ ◆❤â♠ ❤ú✉ ❤↕♥ ✣÷❛ r❛ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ ♥❤â♠ ❤ú✉ ❤↕♥✱ ♠ët sè t➼♥❤ ❝❤➜t✱ ✤à♥❤ ❧➼ ✈➔ ❤➺ q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ♥❤â♠ ❤ú✉ ❤↕♥✳ ❈❤÷ì♥❣ ✸✿ ❇✐➸✉ ❞✐➵♥ ♥❤â♠ ❤ú✉ ❤↕♥ ❚r➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ ❜✐➸✉ ❞✐➵♥ ♥❤â♠ ❤ú✉ ❤↕♥✱ ♠ët ✈➔✐ ✤à♥❤ ❧➼ ❤➺ q✉↔ ❧✐➯♥ q✉❛♥ ✤➳♥ ❜✐➸✉ ❞✐➵♥ ♥❤â♠ ❤ú✉ ❤↕♥✳ ▼➦❝ ❞ò ✤➣ ❝â ❝è ❣➢♥❣ ♥❤✐➲✉ s♦♥❣ ❞♦ ❝❤÷❛ ❝â ❦✐♥❤ ♥❣❤✐➺♠✱ t❤í✐ ❣✐❛♥ ❝ơ♥❣ ♥❤÷ ♥➠♥❣ ❧ü❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ✸ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ rt ữủ sỹ õ ỵ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✹ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ◆❤â♠ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ X ❧➔ t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣✱ tr➯♥ X tr❛♥❣ ❜à ♣❤➨♣ t♦→♥ ❤❛✐ ♥❣ỉ✐ (.)✱ X ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥❤â♠ ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥ ✿ ✭✐✮ ❱ỵ✐ ♠å✐ x, y, z ∈ X t õ (x.y).z = x.(y.z) ỗ t e ∈ X s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x ∈ X, e.x = x.e = x✳ ✭✐✐✐✮ ❱ỵ✐ ♠å✐ x ∈ X tỗ t x X : x.x = x x = e ú ỵ ã e tọ (ii) ❣å✐ ❧➔ ✤ì♥ ✈à ❝õ❛ ♥❤â♠✱ ♣❤➛♥ tû x t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (iii) ✤÷đ❝ ❣å✐ ❧➔ ♣❤➛♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ x ❦➼ ❤✐➺✉ ❧➔ x−1✳ • ◆❤â♠ X ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ ❆❜❡❧ ♥➳✉ ♣❤➨♣ t♦→♥ ❤❛✐ ♥❣æ✐ tr♦♥❣ X ❝â t➼♥❤ ❝❤➜t ❣✐❛♦ ❤♦→♥✱ tù❝ ❧➔ ✈ỵ✐ ♠å✐ x, y ∈ X : x.y = y.x✳ ❚➼♥❤ ❝❤➜t✿ ❈❤♦ X ❧➔ ♠ët ♥❤â♠✳ ❑❤✐ ✤â✿ ✺ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ❙✉② r❛ t❛ ❝â ✿ det(B − λE) = det(T AT −1 − λE) = det(T (A − λE)T −1 = detT det(A − λE)detT −1 = det(A − λE) n ❉♦ α1 = Aii ❝❤ù♥❣ tä T r(a) ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ❝ì sð (ei)ni=1✳ ❍ì♥ i=1 ♥ú❛ →♣ ❞ư♥❣ ✤à♥❤ ❧➼ ❱✐✲❡t ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ det(A − λE) = ✈ỵ✐ ➞♥ λ t❛ ❝â T r(a) = λ1 + λ2 + + λn ✭❝→❝ λi ❦❤æ♥❣ ♥❤➜t t❤✐➳t ♣❤↔✐ ♣❤➙♥ ❜✐➺t ✮✳ ❍❛② T r(a) ❝â ❣✐→ trà ❜➡♥❣ tê♥❣ ❝→❝ ❣✐→ trà r✐➯♥❣ ❦➸ ❝↔ ❜ë✐✳ ✣➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ✣à♥❤ ♥❣❤➽❛ ✸✳✺✳ ●✐↔ sû ϕ : G → GL(V ) ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ t✉②➳♥ t➼♥❤ ❝õ❛ ♥❤â♠ G tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈➨❝tì V ✳ ❍➔♠ sè χϕ : G → C s −→ T r(ϕs ) ✤÷đ❝ ❣å✐ ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ϕ✳ ▼➺♥❤ ✤➲ ✸✳✸✳ ◆➳✉ χ ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ ♠ët ❜✐➸✉ ❞✐➵♥ ϕ ❝â ❝➜♣ ♥ t❤➻✿ ✭✐✮ χ(e) = n✳ ✭✐✐✮ χ(s−1 ) = χ(s)✱ ✈ỵ✐ ♠å✐ s ∈ G✳ ✭✐✐✐✮ χ(tst−1 ) = χ(s)✱ ✈ỵ✐ ♠å✐ s, t ∈ G✳ Ð ✤➙② ❡ ❧➔ ✤ì♥ ✈à ❝õ❛ ●✳ z¯ ❝❤➾ ❧✐➯♥ ❤đ♣ ♣❤ù❝ ❝õ❛ z ∈ C✳ ❈❤ù♥❣ ♠✐♥❤✳ i) ❚❛ ❝â ✿ ϕe = idV s✉② r❛ χ(e) = T r(ϕe) = T r(idV ) = n ✭❱➻ dimV = n✮ ✹✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ❉♦ |G| < ∞✱ s ❝â ❝➜♣ ❤ú✉ ❤↕♥ tr♦♥❣ G✱ ♥➯♥ ϕs ❝â ❝➜♣ ❤ú✉ ❤↕♥ tr♦♥❣ GL(V )✳ ❉♦ K = C ♥➯♥ t❛ ❝â ii) det(A − λE) = (−1)n λn + (−1)n−1 α1 λn−1 + + λ0 = α(λ − λ1 )(λ − λ2 ) (λ − λn ) ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä✱ ❝→❝ ❣✐→ trà r✐➯♥❣ λ1, , λn ❝õ❛ ϕs ❝â ❝➜♣ ❤ú✉ ❤↕♥ tr♦♥❣ ♥❤â♠ ♥❤➙♥ C\{0}✳ ◆â✐ r✐➯♥❣ |λi| = s✉② r❛ λ¯i = (λi)−1 ✈ỵ✐ ♠å✐ i ∈ N ✳ ❑❤✐ ✤â t❛ ❝â ✿ n χ(s) = T r(ϕs ) = ¯i = λ n −1 (λi )−1 = T r(ϕ−1 s ) = T r(ϕs−1 ) = (s ) i=1 i=1 rữợ t t ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❤❛✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ t✉②➳♥ t➼♥❤ ❝õ❛ V ❧➔ a, b ❜➜t ❦➻ ❝â ❝→❝ ♠❛ tr➟♥ t÷ì♥❣ ù♥❣ ❧➔ A = (Aij ); B = (Bij ) tr♦♥❣ ❝ì sð (ei)ni=1 t❤➻ T r(ab) = T r(ba)✳ ❚❤➟t ✈➟②✱ ✤➦t n C = AB = Cij ; Cij = Aik Bkj , i, j = 1, n k=1 n C = BA = Cij ; Cij = Bik Akj , j = 1, n k=1 ❑❤✐ ✤â t❛ ❝â n T r(ab) = Cii = ( k = 1n Aik Bki ) i=1 n n = n Bik Aki = i=1 k=1 Cii = T r(ba) i=1 ❇➡♥❣ ❝→❝❤ ✤➦t u = ts, v = t−1 t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✸✳✹✳ ●✐↔ sû χϕ ✈➔ χψ ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ ❜✐➸✉ ❞✐➵♥ ϕ : ✹✾ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ✈➔ ψ : G → GL(W ) t÷ì♥❣ ù♥❣✳ ❑❤✐ ✤â✿ ✭✐✮ ✣➦❝ tr÷♥❣ χ⊕ ❝õ❛ ❜✐➸✉ ❞✐➵♥ tê♥❣ trü❝ t✐➳♣ ϕ ⊕ ψ ❜➡♥❣ χϕ + χψ ✳ ✭✐✐✮ ✣➦❝ tr÷♥❣ χ⊗ ❝õ❛ ❜✐➸✉ ❞✐➵♥ t➼❝❤ t❡♥①♦ ϕ ⊗ ψ ❜➡♥❣ χϕ.χψ ✳ G → GL(V ) ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû tr♦♥❣ V ❝â ❝ì sð ❧➔ (ei)ni=1✱ tr♦♥❣ W ❝â ❝ì sð ❧➔ (ξi)ni=1✳ ❑❤✐ ✤â V ⊕ W ❝â ❝ì sð ❧➔ (e1, e2, , en, ξ1, ξ2, , ξn)✳ ✣➦t f = ϕ⊕ψ : G → GL(V ⊕W ), fs (v, w) = (ϕs (v), ϕs (w)) ✈ỵ✐ ♠å✐ v ∈ V ❀ ✈ỵ✐ ♠å✐ w ∈ W ✈ỵ✐ ♠å✐ s ∈ G✳ ❑❤✐ ✤â ♠❛ tr➟♥ ❝õ❛ fs ❧➔ ⊕s = φs 0 ψs ❱ỵ✐ φs ❧➔ ♠❛ tr➟♥ ❝õ❛ ϕ tr♦♥❣ ❝ì sð (ei)ni=1✱ ψs ❧➔ ♠❛ tr➟♥ ❝õ❛ ψ tr♦♥❣ ❝ì sð ❝õ❛ (ξi)ni=1✳ ❑❤✐ ✤â T r(⊕s) = T r(ϕs) + T r(ψs) ❤❛② χ⊕(s) = χϕ(s) + χψ (s)✳ ✭✐✐✮ ●✐↔ sû φs = (φij (s)); ψs = (ψij (s))✳ ❚❛ ❝â χϕ (s) = φii (s); χψ (s) = ψkk (s) ✈ỵ✐ ♠å✐ s ∈ G✳ i k ❑❤✐ ✤â χ⊗ (s) = φii (s)ψkk (s) i,k = φii (s) i ψkk (s) k = χϕ (s)χψ (s) ✈ỵ✐ ♠å✐ s ∈ G✳ ✺✵ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ✸✳✸ ❇ê ✤➲ ❙❝❤✉r ❈❤♦ G ❧➔ ♥❤â♠ ❤ú✉ ❤↕♥✱ V, W ❧➔ K ✲ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì✳ ●✐↔ sû ϕ : G → GL(V ) ✈➔ ψ : G → GL(W ) ❧➔ ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝õ❛ G✳ ❈❤♦ f : V → W ❧➔ ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ s❛♦ ❝❤♦ f ϕs = ψs f ✱ ✈ỵ✐ ♠å✐ s ∈ G✳ ◆â✐ ❝→❝❤ ❦❤→❝ f ❧➔ ởt ỗ C [G] ổ õ ỵ ổ ✈ỵ✐ ♥❤❛✉ t❤➻ f = θ ✳ ✷✳ ◆➳✉ V = W ✈➔ ϕ = ψ t❤➻ f ❧➔ ♠ët ♣❤➨♣ ✈à tü tù❝ ❧➔ f ✈ỵ✐ sè ♣❤ù❝ λ ♥➔♦ ✤â✳ = λidV ❈❤ù♥❣ ♠✐♥❤✳ ✶✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ♥➔② t❛ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ ❣✐→♥ t✐➳♣ tù❝ ❧➔ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ ♠å✐ f = t❤➻ f ♣❤↔✐ ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû f = 0✳ ✣➦t V = Kerf ⊂ V ✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ s ∈ G ✈➔ ♠å✐ x ∈ V t❛ ❝â✿ f ϕs (x) = ψs f (x) = ψs (0) = ✣✐➲✉ ✤â ❝❤ù♥❣ tä ϕs(x) ∈ V ✳ ❱➔ ❝â ♥❣❤➽❛ ❧➔ V ữợ t t ❞♦ ϕ ❧➔ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝õ❛ G ✈➔ f ♥➯♥ V = ✭♥➳✉ V = V t❤➻ f = 0✮✳ ❉♦ ✤â f ❧➔ ✤ì♥ ❝➜✉✳ ❚÷ì♥❣ tü t❛ ✤➦t W = Imf ⊂ W ✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ s ∈ G ✈➔ ♠å✐ y ∈ W t❤➻ ψs (y) = ψs f (x) = f (ψs (x)) ⊂ Imf = W ✳ ❉♦ ✤â W ữợ t ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝õ❛ G ✈➔ f = ♥➯♥ W = W ✳ ❉♦ ✤â✱ f ❧➔ t♦➔♥ ❝➜✉✳ ❱➟② f ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ✺✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ❉♦ ✈➟② ϕ ✈➔ ψ ❦❤ỉ♥❣ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥❤❛✉ t❤➻ f = 0✳ ✷✳ ❉♦ C ❧➔ ♠ët tr÷í♥❣ ✤â♥❣ ✤↕✐ sè tù❝ ❧➔ ✤❛ t❤ù❝ ♠ët ➞♥ x õ ữỡ ợ số tr C t➼❝❤ ✤÷đ❝ t❤➔♥❤ t➼❝❤ ♥❤➙♥ tû t✉②➳♥ t➼♥❤✳ ❉♦ ✤â ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ ❧✉ỉ♥ ❝â ♥❣❤✐➺♠ tr♦♥❣ C s✉② r❛ f ❧✉æ♥ ❝â ❣✐→ trà r✐➯♥❣✳ ❑❤æ♥❣ ❣✐↔♠ t➼♥❤ tê♥❣ q✉→t t❛ ❣✐↔ sû λ ❧➔ ♠ët ❣✐→ trà r✐➯♥❣ ❜➜t ❦➻ ❝õ❛ f ✳ ❑❤✐ ✤â t❛ ✤➦t f = f − λidV ✳ ❱ỵ✐ ♠å✐ x ∈ V t❛ ❝â (ψ, f )(x) = ψs [f − λidV (x)] = ψs f (x) − λψs (x) = f ψs (x) − λψs (x) = (f λidV )ϕs (x) = f ϕs (x) ✣✐➲✉ ✤â ❝❤ù♥❣ tä f ϕs = ψsf ✈ỵ✐ ♠å✐ s ∈ K ✳ ▼➦t ❦❤→❝ Kerf = x ∈ V : f (x) = = {x ∈ V : (f − λidV (x))(x) = 0} = {x ∈ V : f (x) − λx = 0} = {x ∈ V : f (x) = λx} = {0} ❱➻ t❤➳ f ❦❤æ♥❣ t❤➸ ❧➔ ♠ët ✤➥♥❣ ❝➜✉ t✉②➳♥ t➼♥❤ ♥➯♥ t❤❡♦ ♣❤➛♥ ✭✶✮ t❛ ❝â f = tù❝ f = λidV ✳ ✸✳✹ ✣➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❛✳ ổ ữợ ổ F (G, C) ❝â ❤➔♠ ♣❤ù❝ tr➯♥ G ❝â ♠ët ✺✷ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ♣❤ù❝ ✈ỵ✐ ♣❤➨♣ ❝ë♥❣ ✈➔ ♣❤➨♣ ♥❤➙♥ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ ❝→❝ ❤➔♠ ❝ư t❤➸ ❧➔ ✿ (α + β)(s) = α(s) + β(s) (cα)(s) = cα(s) ✈ỵ✐ α, β ∈ F (G, C), s ∈ G, c ∈ C✳ α(t)β(t) ①→❝ ✤à♥❤ ♠ët ❚r➯♥ F (G, C) ❜✐➸✉ t❤ù❝ ✿ α, β = |G| tG t ổ ữợ ự t ợ α, β, γ ∈ F (G, C), λ ∈ C t❛ ❝â ✿ ❚✐➯♥ ✤➲ ✭✐✮ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➻ ✿ 1 α, β = α(t)β(t) = β(t)α(t) = β, α ✳ |G| t∈G |G| t∈G ❚✐➯♥ ✤➲ ✭✐✐✮ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➻✿ α + β, γ = |G| = |G| = |G| (α + β)(t)γ(t) t∈G (α(t) + β(t))γ(t) t∈G α(t)γ(t) + t∈G |G| β(t)γ(t) t∈G = α, β + β, γ ❚✐➯♥ ✤➲ ✭✐✐✐✮ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➻✿ λα, β = = |G| |G| ✺✸ (λα)(t)β(t) t∈G λα(t)β(t) t∈G ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ = λ |G| α(t)β(t) t∈G = λ α, β ❚✐➯♥ ✤➲ ✭✐✈✮ ❞÷đ❝ t❤ä❛ ♠➣♥ ✈➻✿ 1 α(t)α(t) = |α(t)|2 |G| t∈G |G| t∈G |α(t)|2 = ⇔ α(t) = α, α = ⇔ |G| t∈G ❱➟② ❜✐➸✉ t❤ù❝ α, β ①→❝ ✤à♥❤ ♠ët t➼❝❤ ổ ữợ , = trữ t q ✣à♥❤ ♥❣❤➽❛ ✸✳✻✳ ✣➦❝ tr÷♥❣ ❝õ❛ ♠ët ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ✤÷đ❝ ❣å✐ ❧➔ ✤➦❝ tr÷♥❣ ❜➜t ❦❤↔ q✉②✳ ❚➼♥❤ ❝❤➜t✿ ❈→❝ ✤➦❝ tr÷♥❣ ❜➜t ❦❤↔ q✉② ❧➟♣ t❤➔♥❤ ởt trỹ ỵ χ ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❦❤ỉ♥❣ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥❤❛✉ t❤➻ ✿ ✭✐✮ ✭✐✐✮ χ, χ = 1✳ χ, χ = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ϕ : G → GL(V ) ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ ❝õ❛ G✳ ❑❤✐ ✤â✱ ♥❣÷í✐ t❛ ❝â t❤➸ ởt t ổ ữợ , tr V t ❜✐➳♥ ✤è✐ ✈ỵ✐ ϕ✳ ❚ù❝ ❧➔ t❤ä❛ ♠➣♥ ❤➺ t❤ù❝ ✿ ϕs (x), ϕs (y) = x, y ✈ỵ✐ ♠å✐ s ∈ G, x, y ∈ V ✳ ❚❤➟t ✈➟②✱ ♥➳✉ , ❝❤÷❛ ❝â t➼♥❤ ❝❤➜t tr➯♥ t❤➻ t❛ t t ổ ữợ ợ x, y = ϕt(x), ϕt(y) ✈ỵ✐ ♠å✐ x, y ∈ V ✳ t∈G ✺✹ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ❑❤✐ ✤â t❛ ❞➵ ❞➔♥❣ ❝â ✤÷đ❝ < , >1 ❜➜t ❜✐➳♥ ✈ỵ✐ ϕ✳ ❚❤➟t ✈➟②✿ < ϕs (x), ϕs (y) >1 = < ϕt ϕs (x), ϕt ϕs (y) > t∈G = < ϕts (x), ϕts (y) > t∈G = < ϕu (x), ϕu (y) > u∈G = < x, y >1 ❑❤✐ ✤â✱ tr♦♥❣ ♠å✐ ❝ì sð trü❝ ❝❤✉➞♥ , ❝õ❛ V ✱ ♠❛ tr➟♥ φt = (φij (t)) ❝õ❛ ϕt ❧➔ ♠❛ tr➟♥ ✉♥✐t❛ tù❝ ❧➔ ✿ φt φt = φt φt = I ✈ỵ✐ φt = (φij ) tr♦♥❣ ✤â (φij (t)) = (φji (t)) ▲ó❝ ✤â✱ t❛ ❝â (φij (t−1)) = (φij (t))✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧➼✳ ✐✮ ●✐↔ sû χ ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ϕ✱ ✤÷đ❝ ❝❤♦ tr♦♥❣ ♠ët ❝ì sð ♥➔♦ ✤➜② ❝õ❛ ♠❛ tr➟♥ ❯♥✐t❛ φt = φij (t), t ∈ G✳ ❑❤✐ ✤â χ(t) = φii(t)✳ i ❚❛ ❝â χ, χ = |G| = ( φii (t)φkk (t) = t,i,k |G| φkk (t−1 φii (t) t,i,k δik )/n i,k = n =1 n ✐✐✮ ❈❤♦ ❤❛✐ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ϕ, ψ ❝õ❛ G ❦❤ỉ♥❣ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥❤❛✉✳ ●å✐ χ, χ ❧➛♥ ❧÷đt ❧➔ ❤❛✐ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ φ ✈➔ ψ ✺✺ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ✤÷đ❝ ❝❤♦ tr♦♥❣ ♠ët ❝ì sð ♥➔♦ ✤â ❝õ❛ ♠❛ tr➟♥ ❯♥✐t❛ ✿ φt = (φij (t)); ψt = (ψij (t)) ✈ỵ✐ ♠å✐ t ∈ G✳ ❑❤✐ ✤â t❛ ❝â χ(t) = φii(t); χ (t) = ψji(t) i j ❙✉② r❛ χ, χ = = |G| φii (t)ψjj (t) i,j |G| ψij (t−1 )φii (t) = i,j ●✐↔ sû ❱ ❧➔ ♠ët ●✲❦❤æ♥❣ ợ trữ sỷ ữủ ♣❤➙♥ t➼❝❤ t❤➔♥❤ ❝→❝ ●✲❦❤æ♥❣ ❣✐❛♥ ❜➜t ❦❤↔ q✉② V = W1 ⊕ W2 ⊕ ⊕ Wk ✳ ❑❤✐ ✤â✱ ♥➳✉ ❲ ❧➔ ♠ët ● ✲❦❤æ♥❣ ❣✐❛♥ ❜➜t ❦❤↔ q ợ trữ t số Wi ❝➜✉ ✈ỵ✐ ❲ ❧➔ ❜➡♥❣ α, χ ✳ ❍➺ q✉↔ ✸✳✶✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû χi ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ Wi ✈ỵ✐ ♠å✐ i = 1, k ❦❤✐ ✤â t❤❡♦ ♠➺♥❤ ✤➲ ✸✳✸ ✈➲ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ tê♥❣ trü❝ t✐➳♣ t❛ ❝â α = χ1 + χ2 + + χk ✳ ❉♦ ✤â α, χ = χ1 + χ2 + + χk , χ = χ1 , χ + + χk , χ ▼➦t ❦❤→❝ t❤❡♦ ✤à♥❤ ❧➼ tr➯♥ t❛ ❝â ✿ χi , χ = 1 0 ❱➟② α, χ ♥➳✉ Wi ✤➥♥❣ ❝➜✉ ✈ỵ✐ ❲ ♥➳✉ Wi ❦❤ỉ♥❣ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ❲ ❝❤➼♥❤ ❧➔ sè ❝→❝ Wi ✤➥♥❣ ❝➜✉ ✈ỵ✐ W ✳ ✺✻ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✣à♥❤ ỵ ổ tự ❚♦→♥ ✣➦❝ tr÷♥❣ rG ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② G ✤÷đ❝ ❝❤♦ ❜ð✐ rG (s) = |G| 0 ♥➳✉ s = e ♥➳✉ s = e Ð ✤➙② ❡ ❧➔ ✤ì♥ ✈à ❝õ❛ ●✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ ♠ët ❝ì sð ❝õ❛ C(G) ❧➔ (t)t∈G✳ ⑩♥❤ ①↕ ϕs õ t ữ s s(t) = st ợ ♠å✐ s ∈ G ✐✮ ◆➳✉ s = e t❤➻ st = t ✈ỵ✐ ♠å✐ t ∈ G ♠å✐ s G\{e} ữợ tr t❛ ✤÷đ❝ ♠❛ tr➟♥ Aϕ = (aij )n✳ ❙✉② r❛ ❝→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ✤➲✉ ❜➡♥❣ ✵✳ ❚ù❝ ❧➔ rG(s) = T r(ϕs) = 0✱ ✈ỵ✐ ♠å✐ s = e✳ ✐✐✮ ◆➳✉ s = e t❤➻ st = t✳ ❑❤✐ ✤â✱ ❝→❝ ♣❤➛♥ tû tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝õ❛ ♠❛ tr➟♥ ϕs ✤è✐ ✈ỵ✐ ❝ì sð ❝õ❛ G ✤➲✉ ❜➡♥❣ ✶ ♥➯♥ t❛ ❝â s rG (s) = T r(ϕs ) = dimC[G] = |G| ✳ ❍➺ q✉↔ ✸✳✷✳ ▼é✐ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ✤➲✉ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② ✈ỵ✐ sè ❜ë✐ ❜➡♥❣ ❝➜♣ ❝õ❛ ♥â✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû χ ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ G✲ ❦❤æ♥❣ ❣✐❛♥ ❜➜t ❦❤↔ q✉② W ❝â ❝➜♣ n✳ ❚❤❡♦ ❤➺ q✉↔ ✸✳✶✱ sè ❧➛♥ W ①✉➜t ❤✐➺♥ tr♦♥❣ C[G] ❜➡♥❣ 1 rG , χ = rG (t)χt = |G|χ(t) = n✳ |G| |G| t∈G ●✐↔ sû W1, W2, , Wh ❧➔ t➜t ❝↔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝õ❛ ● ❜➜t ❦❤↔ q✉②✱ ✤ỉ✐ ♠ët ❦❤ỉ♥❣ ✤➥♥❣ ❝➜✉ ✈ỵ✐ ♥❤❛✉✱ ❝â ❝→❝ ✤➦❝ tr÷♥❣ t÷ì♥❣ ù♥❣ ❧➔ χ1, χ2, , χh ✈➔ ❝➜♣ t÷ì♥❣ ù♥❣ ❝õ❛ ♥â ❧➔ n1, n2, , nh✳ ❑❤✐ ✤â✿ ❍➺ q✉↔ ✸✳✸✳ ✺✼ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ✶✳ n21 + + n2h = |G|✳ ✷✳ h i=1 ni χi (s) = 0, ∀s ∈ G\{e}✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â C[G] = n1W1 ⊕ n2W2 ⊕ ⊕ nhWh✳ ❑❤✐ ✤â t❛ ❝â h rG (s) = ni χi (s) ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ❝❤➼♥❤ q✉② ❝õ❛ G✳ i=1 ❚❤❡♦ ✤à♥❤ ❧➼ ✸✳✶ t❛ ❝â✿ • • ◆➳✉ s❂❡ t❤➻ rG(s) = h h ni χi (s) = i=1 i=1 ◆➳✉ s = e t❤➻ rG(s) = h i=1 n2i = |G|✳ ni χi (s) = 0✳ ✸✳✺ ❙è ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❍➔♠ f : G → C ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❤➔♠ ❧ỵ♣ tr➯♥ G ♥➳✉ f (tst−1) = f (s) ✈ỵ✐ ♠å✐ s, t ∈ G✳ ❑❤✐ ✤â t➟♣ ❝→❝ ❤➔♠ ❧ỵ♣ tr➯♥ G ❦➼ ❤✐➺✉ ❧➔ RC(G) ❧➔♠ t❤➔♥❤ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ❝♦♥ ❝õ❛ F (G, C)✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✼✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ f, g ∈ RC(G) t❛ ❝â (f + g)(tst−1 ) = f (tst−1 ) + g(tst−1 ) = f (s) + g(s) = (f + g)(s) ✈ỵ✐ ♠å✐ s, t ∈ G, λ ∈ C✳ ✣✐➲✉ ✤â ❝❤ù♥❣ tä RC(G) ✤â♥❣ ❦➼♥ ✤è✐ ợ ổ ữợ tr F (G, C)✳ (λf )(tst−1 ) = λf (s) ✺✽ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ●✐↔ sû ❢ ❧➔ ♠ët ❤➔♠ ❧ỵ♣ tr➯♥ ● ✈➔ ϕ : G → GL(V ) ❧➔ ♠ët ❜✐➸✉ t q ợ trữ χ✳ ❑❤✐ ✤â ♣❤➨♣ ❜✐➳♥ ✤ê✐ t✉②➳♥ t➼♥❤ ϕf = f (t)ϕt : V → V ❧➔ ♠ët ♣❤➨♣ ✈à tü t❤❡♦ t➾ ❧➺ ❇ê ✤➲ ✸✳✶✳ t∈G λ= |G| f, χ n ✳ ❚r♦♥❣ ✤â χ ❧➔ ❤➔♠ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ (χ)(s) ¯ = χ(s) ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ s ∈ G t❛ ❝â ✿ ϕ−1 s ϕf ϕs = f (t)ϕ−1 s ϕt ϕs t∈G = ft ϕs−1 ts t∈G ✣➦t u = s−1ts t❤➻ f (t) = f (sus−1) = f (u) ✭❞♦ f ❧➔ ♠ët ❤➔♠ ❧ỵ♣ tr➯♥ f (u)ϕu s✉② r❛ ϕf ϕs = ϕs ϕu = ϕf G✮ ❝❤♦ ♥➯♥ ϕ−1 s ϕf ϕs = u∈G ❚❤❡♦ ❜ê ✤➲ ❙❝❤✉r ϕf = λidV ♥➯♥ t❛ ❝â nλ = T r(λidV ) = T r(ϕf ) = |G| f, χ ✳ f (t)T r(ϕt ) = f (t)χ(t) = |G| f, χ¯ ✳ ❙✉② r❛ λ = n t∈G ●å✐ χ1, , χh ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ t➜t ❝↔ ♥❤ú♥❣ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ✤æ✐ ♠ët ❦❤æ♥❣ ✤➥♥❣ ❝➜✉ ❝õ❛ ●✳ ❑❤✐ ✤â χ1, , χh ❧➟♣ ♥➯♥ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ RC(G) ❝õ❛ ❝→❝ ❤➔♠ ❧ỵ♣ tr➯♥ ● ỵ ự t❛ ❝â χ1, , χh ❧➔ ♠ët ❤➺ trü❝ ❝❤✉➞♥ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❝❤ó♥❣ ❧➔ ♠ët ❤➺ s✐♥❤ ❝õ❛ RC(G) ▼✉è♥ ✈➟② t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ f ∈ RC(G) ✈➔ f trü❝ ❣✐❛♦ ✈ỵ✐ ♠å✐ χi t❤➻ f = 0✳ ◆➳✉ ϕ ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝õ❛ G t❤➻ t❤❡♦ ❜ê ✤➲ tr➯♥ t❛ ❝â ϕf = f (t)ϕt = 0✱ ✈➻ f trü❝ ❣✐❛♦ ✈ỵ✐ ♠å✐ χi ♠➔ ♠ët tr♦♥❣ ❝→❝ ❤➔♠ t∈G χi ♣❤↔✐ ❧➔ ✤➦❝ tr÷♥❣ ❝õ❛ ϕ ✳ ❱➻ ♠é✐ ❜✐➸✉ ❞✐➵♥ ✤➲✉ ❧➔ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ❝❤♦ ♥➯♥ ϕf = ✈ỵ✐ ♠å✐ ❜✐➸✉ ❞✐➵♥ ϕ ✳ ✺✾ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ❱➟② ✈ỵ✐ e ❧➔ ✤ì♥ ✈à ❝õ❛ G t❛ ❝â ✿ = ϕf (e) = f (t)ϕt(e) = f (t)t✳ t∈G t∈G ❚ø ✤â f (t) = 0✱ ✈ỵ✐ ♠å✐ t ∈ G✳ ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä χ1, χ2, , χh ❧➟♣ t❤➔♥❤ ♠ët ❤➺ trü❝ ❝❤✉➞♥ ❝õ❛ RC (G)✳ ❙è ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ✤æ✐ ♠ët ❦❤æ♥❣ ✤➥♥❣ ❝➜✉ ❝õ❛ số ợ ủ ỵ ✸✳✺✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû D1, , Dk ❧➔ t➜t ❝↔ ❝→❝ ❧ỵ♣ ❧✐➯♥ ❤đ♣ tr♦♥❣ G✳ ❍➔♠ f : G → C ❧➔ ♠ët ❤➔♠ ❧ỵ♣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f ❧➔ ♠ët ❤➡♥❣ sè tr➯♥ ♠é✐ ❧ỵ♣ ❧✐➯♥ ❤ñ♣ Di(i = 1, , k)✳ ❈→❝ ❤➡♥❣ sè ♣❤ù❝ õ t tũ ỵ t dimRC(G) = k✳ ▼➦t ❦❤→❝ t❤❡♦ ✤à♥❤ ❧➼ tr➯♥ dimRC(G) ❜➡♥❣ sè ❝→❝ ❜✐➸✉ ❞✐➵♥ ❜➜t ❦❤↔ q✉② ✤æ✐ ♠ët ❦❤æ♥❣ ✤➥♥❣ ❝➜✉ ❝õ❛ G✳ ✻✵ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✲ ❑✸✾❆ ❚♦→♥ ❑➳t ❧✉➟♥ ❚r♦♥❣ ❦❤â❛ tr õ ỳ ỗ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ s❛✉✿ ◆❤â♠ ❤ú✉ ❤↕♥✱ ♠ët sè ♥❤â♠ ❤ú✉ ❤↕♥ ♥❤÷ ♥❤â♠ ✤è✐ ①ù♥❣✱ ♥❤â♠ t❤❛② ♣❤✐➯♥✱ ♥❤â♠ ❆❜❡❧ ❤ú✉ ❤↕♥✱ ♥❤â♠ ◗✉❛t❡r♥✐♦♥ ❝ị♥❣ ✈ỵ✐ ❧➼ t❤✉②➳t ❜✐➸✉ ❞✐➵♥ ♥❤â♠ ❤ú✉ ❤↕♥✳ ▼➦❝ ❞ò ✤➣ r➜t ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝õ❛ ❜↔♥ t❤➙♥ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❱➻ ✈➟② ❡♠ r➜t ♠♦♥❣ ữủ ỳ ỵ õ õ tứ t ổ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ▼ët ❧➛♥ ♥ú❛ ❡♠ ①✐♥ ✤÷đ❝ ❝↔♠ ì♥ sỹ ữợ ú ù t t ổ ✲ ❚✐➳♥ s➽ ◆❣✉②➵♥ ❚❤à ❑✐➲✉ ◆❣❛ ✱ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➣ ❣✐ó♣ ❡♠ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✻✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ❚ü ❈÷í♥❣ ✱ ●✐→♦ tr➻♥❤ ✤↕✐ sè ❤✐➺♥ ✤↕✐✱ ◆❳❇ ✣❍◗● ❍➔ ◆ë✐✱ ✷✵✵✸ ✳ ❬✷❪ ◆❣✉②➵♥ ❍ú✉ ❱✐➺t ❍÷♥❣✱ ✣↕✐ sè ✤↕✐ ❝÷ì♥❣✱ ◆❳❇ ●❉✱ ✶✾✾✽✳ ❬✸❪ ❍♦➔♥❣ ❳✉➙♥ ❙➼♥❤✱ ✣↕✐ sè ✤↕✐ ❝÷ì♥❣ ✱ ◆❳❇ ●❉✱ ✷✵✵✼✳ ❬✹❪ ❉÷ì♥❣ ◗✉è❝ ❱✐➺t✱ ▼ët sè ❝➜✉ tró❝ ❝ì ❜↔♥ ❝õ❛ ✣↕✐ sè ❤✐➺♥ ✤↕✐ ✱ ◆❳❇ ✣❍❙P ❍➔ ◆ë✐✱ ✷✵✵✽✳ ✻✷