1. No category

‫ﻤﺠﻠﺔ ﺠﺎﻤﻌﺔ ﺩﻤﺸﻕ ﻟﻠﻌﻠﻭﻡ ﺍﻷﺴﺎﺴﻴﺔ ـ ﺍﻟﻤﺠﻠﺩ )‪ (١٦‬ـ ﺍﻟﻌﺩﺩ ﺍﻷﻭل ـ ‪٢٠٠٠‬‬
‫ﻤﻨﻅﻡ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل‪ ،‬ﻭﺫﺍﺕ ﺍﻟﺭﺘﺒﺔ‬
‫ﺍﻟﻔﺭﺩﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ )‪ ،GL(q,p‬ﺤﻴﺙ ‪ p‬ﻭ ‪ q‬ﻋﺩﺩﺍﻥ ﺃﻭﻟﻴﺎﻥ‬
‫ﺩ‪ .‬ﺍﺴﻜﻨﺩﺭ ﻋﻠﻲ ﻭ ﺴﻠﻭﻯ ﻴﻌﻘﻭﺏ‬
‫ﻗﺴﻡ ﺍﻟﺭﻴﺎﻀﻴﺎﺕ ـ ﻜﻠﻴﺔ ﺍﻟﻌﻠﻭﻡ ـ ﺠﺎﻤﻌﺔ ﺘﺸﺭﻴﻥ ـ ﺍﻟﻼﺫﻗﻴﺔ ـ ﺍﻟﺠﻤﻬﻭﺭﻴﺔ ﺍﻟﻌﺭﺒﻴﺔ ﺍﻟﺴﻭﺭﻴﺔ‬
‫ﺘﺎﺭﻴـﺦ ﺍﻹﻴﺩﺍﻉ ‪١٩٩٨/٠٢/٢٠‬‬
‫ﻗﺒل ﻟﻠﻨﺸـﺭ ﻓﻲ ‪١٩٩٩/٠٥/٢٥‬‬
‫ﺍﻟﻤﻠﺨﺹ‬
‫ﺴﻭﻑ ﻨﺴﺘﻌﺭﺽ ﻓﻲ ﻫﺫﻩ ﺍﻟﺩﺭﺍﺴﺔ ﻤﺴﺄﻟﺔ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ GL(q,p‬ﺤﻴـﺙ ‪q,p‬‬
‫ﻋﺩﺩﺍﻥ ﺃﻭﻟﻴﺎﻥ‪ ،q|(p-١) ،‬ﻭﺍﻟﺘﻲ ﺘﻤﻠﻙ ﺭﺘﺒﺔ ﻤﺎ ‪ ،r‬ﺤﻴﺙ )‪ ، r /| ( P − 1‬ﻤﻥ ﺤﻴﺙ ﺩﺭﺍﺴﺔ ﺯﻤﺭﻫـﺎ ﺍﻟﺠﺯﺌﻴـﺔ‬
‫ﺍﻟﻨﻅﺎﻤﻴﺔ ﻭﺘﺤﺩﻴﺩ ﺭﺘﺒﺔ ﺍﻟﻤﻨﻅﻡ ﻟﻜ ﱟل ﻤﻨﻬﺎ ﻓﻲ ﺍﻟﺤﺎﻟﺔ ﺍﻟﺘﻲ ﻴﻜﻭﻥ ﻓﻴﻬﺎ ‪ r‬ﻋﺩﺩﹰﺍ ﻓﺭﺩﻴﺎﹰ‪ ،‬ﻭﻜﺫﻟﻙ ﺩﺭﺍﺴـﺔ ﻤﺴـﺄﻟﺔ‬
‫ﺘﺭﺍﻓﻘﻬﺎ ﻓﻲ ﺍﻟﺯﻤﺭﺓ )‪.GL(q,p‬‬
‫ﺍﻟﻜﻠﻤﺎﺕ ﺍﻟﻤﻔﺘﺎﺡ‪ :‬ﺯﻤﺭﺓ ـ ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ـ ﻤﻨﻅﻡ ـ ﺯﻤﺭﺓ ﻗﺎﺒﻠﺔ ﻟﻠﺤل‪.‬‬
‫‪١٠١‬‬
… ‫ﻴﻌﻘﻭﺏ ﻭ ﻋﻠﻲ ـ ﺩﺭﺍﺴﺔ ﻤﻨﻅﻡ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻭﺫﺍﺕ ﺍﻟﺘﺭﺒﺔ ﺍﻟﻔﺭﺩﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ‬
Normalijor of Isolating Soluable Subgroups of
Odd Order in the Qroup GL(q, p),
Where p, q are prime sumfers
Salwa YAAQOOP , Dr Eskandar ALI
Mathematics Department-Faculty of Science-Tishreen University-Lattaquia-Syria
Received ٢٠/٠٢/١٩٩٨
Accepted ٢٥/٠٥/١٩٩٩
ABSTRACT
This study represents the problem of the invariant sub-groups of the
maximal solvable invariant sub-groups in a group GL(q,p), with p and q are
primes, q|(p-١), and its order is r with r /| ( P − 1) , as the studying of its
normalizer groups and identification the order of the normalizer for each one
in the case r is an odd order, and also with studying the problem of its
conjunction in a group GL(q,p).
Key Words: Group, Subgroup, Normalizor, Solvable Group.
١٠٢
‫ﻤﺠﻠﺔ ﺠﺎﻤﻌﺔ ﺩﻤﺸﻕ ﻟﻠﻌﻠﻭﻡ ﺍﻷﺴﺎﺴﻴﺔ ـ ﺍﻟﻤﺠﻠﺩ )‪ (١٦‬ـ ﺍﻟﻌﺩﺩ ﺍﻷﻭل ـ ‪٢٠٠٠‬‬
‫ﺘﻌﺎﺭﻴﻑ ﻭﻤﺼﻁﻠﺤﺎﺕ‪:‬‬
‫‪-‬‬
‫⎞‪⎛n‬‬
‫ﻨﺩﻋﻭ ﺍﻟﺤﻘل ⎟ ‪ GF ⎜ p‬ﺤﻴﺙ ‪ p‬ﻋﺩﺩ ﺃﻭﻟﻲ ﻭ ‪ n‬ﻋﺩﺩ ﻁﺒﻴﻌـﻲ‪ ،‬ﺒﺤﻘـل ﻏـﺎﻟﻭﺍ ﻤـﻥ‬
‫⎠ ⎝‬
‫‪n‬‬
‫‪-‬‬
‫ﺍﻟﺭﺘﺒﺔ ‪. p‬‬
‫ﺍﻟﺯﻤﺭﺓ )‪ GL(V‬ﻫﻲ ﺯﻤﺭﺓ ﻤﺼﻔﻭﻓﺎﺕ ﺠﻤﻴﻊ ﺍﻟﻤﺅﺜﺭﺍﺕ ﺍﻟﺨﻁﻴﺔ ﺍﻟﻤﺘﺒﺎﻴﻨﺔ ﻋﻠﻰ ﺍﻟﻔﻀﺎﺀ‬
‫‪.V‬‬
‫‪-‬‬
‫ﻨﺴﻤﻲ ﺍﻟﺯﻤﺭﺓ ‪ H‬ﺍﻟﺠﺯﺌﻴﺔ ﻤﻥ ﺍﻟﺯﻤﺭﺓ )‪ ،GL(V‬ﺯﻤﺭﺓ ﻨﺎﻗﻠﺔ ﻋﻠﻰ ﺍﻟﻔﻀﺎﺀ ‪ V‬ﺇﺫﺍ ﻭﺠـﺩ‬
‫ﻓﻲ ﺍﻟﻔﻀﺎﺀ ‪ V‬ﻓﻀﺎﺀ ﺠﺯﺌﻲ ﻭﺍﺤﺩ ﻋﻠﻰ ﺍﻷﻗـل ﻤﺜـل ‪ ،V١‬ﺒﺤﻴـﺙ‬
‫)ﺍﻟﻔﻀﺎﺀ ﺍﻟﺼﻔﺭﻱ( ﻭﻜﺎﻥ‪:‬‬
‫‪{0} ≠ V1 ≠ V‬‬
‫‪h (V1 ) = V1 ; ∀h ∈ H‬‬
‫ﺃﻤ‪‬ﺎ ﺇﺫﺍ ﻜﺎﻨﺕ ﺍﻟﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴﺔ ‪ H‬ﻻ ﺘﻨﻘل ﻤﻥ ﺍﻟﻔﻀﺎﺀﺍﺕ ﺍﻟﺠﺯﺌﻴﺔ ﻓﻲ ﺍﻟﻔﻀﺎﺀ ‪ V‬ﺴـﻭﻯ‬
‫ﺍﻟﻔﻀﺎﺌﻴﻥ ﺍﻟﺠﺯﺌﻴﻴﻥ ‪ V‬ﻭ }‪ {0‬ﻋﻨﺩﺌﺫ ﻨﺩﻋﻭﻫﺎ ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ )‪ GL(V‬ﻋﻠﻰ‬
‫ﺍﻟﻔﻀﺎﺀ ‪.V‬‬
‫‪-‬‬
‫ﻨﻘﻭل ﻋﻥ ﺍﻟﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴﺔ ‪ GL(V)⊃H‬ﺇﻨﹼﻬﺎ ﺯﻤﺭﺓ ﻏﻴﺭ ﺃﺼﻠﻴﺔ‪ ،‬ﺇﺫﺍ ﻜﺎﻥ ﺍﻟﻔﻀـﺎﺀ ‪V‬‬
‫⊕‬
‫ﻴﻜﺘﺏ ﻋﻠﻰ ﺸﻜل ﻤﺠﻤﻭﻉ ﻤﺒﺎﺸﺭ ‪ V = ∑ Qv ; v ∈ I‬ﻟﻔﻀﺎﺀﺍﺕ ﺠﺯﺌﻴﺔ ‪، {0} ≠ Qv‬‬
‫ﻭﻜﺎﻥ ‪) Card I>١‬ﺤﻴﺙ ‪ I‬ﻤﺠﻤﻭﻋﺔ ﺠﺯﺌﻴﺔ ﻤﻥ ﻤﺠﻤﻭﻋﺔ ﺍﻷﻋﺩﺍﺩ ﺍﻟﻁﺒﻴﻌﻴﺔ(‪ ،‬ﻭﻜﺎﻥ ﻤﻥ‬
‫ﻥ ‪ g (Qα ) ⊆ Qβ‬ﺤﻴﺙ ‪. I ∋ β‬‬
‫ﺃﺠل ﺃﻱ ‪ I ∋ α‬و ‪ H ∋ g‬ﺃ ‪‬‬
‫ﺃﻤ‪‬ﺎ ﺇﺫﺍ ﻜﺎﻨﺕ ‪ H‬ﻻ ﺘﺤﻘﻕ ﺍﻟﺸﺭﻭﻁ ﺍﻟﺴﺎﺒﻘﺔ ﻓﻨﺩﻋﻭﻫﺎ ﺯﻤﺭﺓ ﺃﺼﻠﻴﺔ‪.‬‬
‫‪-‬‬
‫ﻨﺩﻋﻭ ﺍﻟﺯﻤﺭﺓ ‪ H‬ﺯﻤﺭﺓ ﻗﺎﺒﻠﺔ ﻟﻠﺤل ﺇﺫﺍ ﻭﺠﺩﺕ ﻓﻴﻬﺎ ﺴﻠﺴﻠﺔ ﻤـﻥ ﺍﻟﺯﻤـﺭ ﺍﻟﺠﺯﺌﻴـﺔ ‪Hi‬‬
‫ﺒﺤﻴﺙ‪:‬‬
‫‪=H‬‬
‫‪{1} = H 0 ⊆ H 1 ⊆ ... ⊆ H r‬‬
‫ﺤﻴﺙ ‪ H i +1 ∆H i‬ﺘﻌﺭﻴﻑ ﺍﻟﺯﻤﺭﺓ ﺍﻟﻨﻅﺎﻤﻴﺔ )ﺃﻱ ﺍﻟﺯﻤﺭﺓ ‪ H i‬ﻨﻅﺎﻤﻴﺔ ﻓﻲ ﺍﻟﺯﻤـﺭﺓ ‪،( H i +1‬‬
‫‪H‬‬
‫ﻭﺍﻟﺯﻤﺭﺓ ﺍﻟﻌﺎﻤﻠﺔ‬
‫‪ i +1‬ﺠﻤﻴﻌﻬﺎ ﺘﺒﺩﻴﻠﻴﺔ‪.‬‬
‫‪Hi‬‬
‫‪١٠٣‬‬
‫ﻴﻌﻘﻭﺏ ﻭ ﻋﻠﻲ ـ ﺩﺭﺍﺴﺔ ﻤﻨﻅﻡ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻭﺫﺍﺕ ﺍﻟﺘﺭﺒﺔ ﺍﻟﻔﺭﺩﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ …‬
‫‪-‬‬
‫ﻼ ﺯﻤﺭﻴﹰﺎ ﻋﻨﺩﺌﺫ ﺴـﻨﺭﻤﺯ ﺒــﹻ ‪Ψ H : H → K‬‬
‫ﺒﻔﺭﺽ ‪ Ψ : G1 → G2‬ﺘﺸﺎﻜ ﹰ‬
‫ﻟﻤﻘﺼﻭﺭ ‪ Ψ‬ﻋﻠﻰ ‪ ،H‬ﺤﻴﺙ ‪ H‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻤﻥ ﺍﻟﺯﻤﺭﺓ ‪ G١‬ﻭ ‪ K‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻤـﻥ‬
‫ﺍﻟﺯﻤﺭﺓ ‪.G٢‬‬
‫‪-‬‬
‫ﺴﻨﺭﻤﺯ ﺒـﹻ ‪ H 1 , H 2‬ﻟﻠﺯﻤﺭﺓ ﺍﻟﻤﻭﻟﹼﺩﺓ ﻤﻥ ﺍﻟﺯﻤﺭﺘﻴﻥ ﺍﻟﺠـﺯﺌﻴﺘﻴﻥ ‪ H١‬ﻭ ‪ H٢‬ﻤـﻥ‬
‫ﺯﻤﺭﺓ ﻤﺎ ‪ ،G‬ﻜﻤﺎ ﺴﻨﺭﻤﺯ ﺒـﹻ ‪ a‬ﻟﻠﺯﻤﺭﺓ ﺍﻟﺩﻭﺭﻴﺔ ﺍﻟﻤﻭﻟﹼﺩﺓ ﺒﻌﻨﺼﺭ ﻭﺍﺤـﺩ ‪ a‬ﻤـﻥ‬
‫ﺍﻟﺯﻤﺭﺓ ‪.G‬‬
‫‬‫‪-‬‬
‫ﻨﺩﻋﻭ ‪ Sx‬ﺒﺯﻤﺭﺓ ﺍﻟﺘﺒﺎﺩﻴل ﺍﻟﻤﻌﺭ‪‬ﻓﺔ ﻋﻠﻰ ﺍﻟﻤﺠﻤﻭﻋﺔ ‪.x‬‬
‫ﺍﻟﺯﻤﺭﺓ )‪ GL(q,p‬ﻫﻲ ﺯﻤﺭﺓ ﺠﻤﻴﻊ ﺍﻟﻤﺼﻔﻭﻓﺎﺕ ﺍﻟﻤﺭﺒﻌﺔ ﺍﻟﻘﻠﻭﺒـﺔ )ﺍﻟﻤﻨﺘﻅﻤـﺔ( ﻤـﻥ‬
‫ﺍﻟﻤﺭﺘﺒﺔ ‪ ،q‬ﻭﺍﻟﻤﺄﺨﻭﺫﺓ ﻋﻨﺎﺼﺭﻫﺎ ﻤﻥ ﺍﻟﺤﻘل )‪ ،GF(P‬ﺤﻴﺙ ‪ p‬ﻭ ‪ q‬ﻋﺩﺩﺍﻥ ﺃﻭﻟﻴـﺎﻥ‪ ،‬ﻭ‬
‫‪ Iq‬ﻤﺼﻔﻭﻓﺔ ﺍﻟﻭﺍﺤﺩﺓ ﻓﻲ ﻫﺫﻩ ﺍﻟﺯﻤﺭﺓ‪.‬‬
‫‪-‬‬
‫ﺍﻟﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴﺔ )‪ SL(q,p‬ﻭﻫﻲ ﺯﻤﺭﺓ ﺠﻤﻴﻊ ﺍﻟﻤﺼﻔﻭﻓﺎﺕ ﻤﻥ ﺍﻟﺯﻤﺭﺓ )‪ GL(q,p‬ﻭﺍﻟﺘﻲ‬
‫‪-‬‬
‫ﻨﺴﻤﻲ ﺍﻟﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴـﺔ }‪ N ( A) = {g ∈ GL(q, p ); gA = Ag‬ﺯﻤـﺭﺓ ﺍﻟﻤـﻨﻅﻡ‬
‫ل ﻤﻨﻬﺎ ﻴﺴﺎﻭﻱ ﺍﻟﻭﺍﺤﺩ‪.‬‬
‫ﻤﻌﻴﻥ ﻜ ﱟ‬
‫ﻟﻠﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴﺔ ‪ A‬ﻓﻲ ﺍﻟﺯﻤﺭﺓ )‪ ،GL(q,p‬ﻭﻫﻲ ﺃﻜﺒﺭ ﺯﻤﺭﺓ ﺠﺯﺌﻴـﺔ ﻓـﻲ ﺍﻟﺯﻤـﺭﺓ‬
‫)‪ GL(q,p‬ﺘﻜﻭﻥ ‪ A‬ﻨﻅﺎﻤﻴﺔ ﻓﻴﻬﺎ‪.‬‬
‫‪-‬‬
‫ﻨﺩﻋﻭ ﺍﻟﻌﻨﺼﺭ ‪ (a, b ) = aba −1b −1‬ﻤﺒﺎﺩل ﺍﻟﻌﻨﺼﺭﻴﻥ ‪ a‬ﻭ ‪ b‬ﻤﻥ ﺍﻟﺯﻤﺭﺓ ‪.A‬‬
‫‪-‬‬
‫ﻟﺘﻜﻥ ‪ A‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻤﺎ‪ ،‬ﻨﺩﻋﻭ ﺍﻟﺯﻤـﺭﺓ ﺍﻟﺠﺯﺌﻴـﺔ ‪A′ = aba −1b −1 ; a, b ∈ A‬‬
‫}‬
‫{‬
‫ﺍﻟﻤﻭﻟﺩﺓ ﺒﻜل ﺍﻟﻤﺒﺎﺩﻻﺕ ﻓﻲ ‪ A‬ﺒﺎﻟﺯﻤﺭﺓ ﺍﻟﻤﺸﺘﻘﺔ ﻟﻠﺯﻤﺭﺓ ‪ A‬ﺃﻭ ﺍﻟﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴﺔ ﻟﻠﻤﺒﺎﺩﻻﺕ‬
‫ﻓﻲ ‪.A‬‬
‫ﺍﻟﻤﺒﺭﻫﻨﺎﺕ‪:‬‬
‫ﻥ ﺍﻟﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻷﺼﻠﻴﺔ ﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ ﺍﻟﺯﻤﺭﺓ )‪ GL(q,p‬ﻫـﻲ‬
‫‪ -١‬ﻤﻌﻠﻭﻡ ﻤﻥ ]‪ [١‬ﺃ ‪‬‬
‫ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ ﺘﻠﻙ ﺍﻟﺯﻤﺭﺓ‪.‬‬
‫‪١٠٤‬‬
‫ﻤﺠﻠﺔ ﺠﺎﻤﻌﺔ ﺩﻤﺸﻕ ﻟﻠﻌﻠﻭﻡ ﺍﻷﺴﺎﺴﻴﺔ ـ ﺍﻟﻤﺠﻠﺩ )‪ (١٦‬ـ ﺍﻟﻌﺩﺩ ﺍﻷﻭل ـ ‪٢٠٠٠‬‬
‫‪ -١‬ﺒﻔﺭﺽ ‪ G‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﻅﻤﻰ ﻋﺎﺯﻟﺔ ﻭﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ GL(q,p‬ﺤﻴﺙ ‪ G‬ﺃﺼﻠﻴﺔ‬
‫ﻭ ‪ p‬ﻭ ‪ q‬ﻋﺩﺩﺍﻥ ﺃﻭﻟﻴﺎﻥ‪ ،‬ﻭﺒﺤﻴﺙ )‪ ،q|(p-١‬ﻭﻟﺘﻜﻥ ‪ F‬ﺯﻤﺭﺓ ﺘﺒﺩﻴﻠﻴﺔ ﻨﻅﺎﻤﻴـﺔ ﻋﻅﻤـﻰ ﻓـﻲ‬
‫ﺍﻟﺯﻤﺭﺓ ‪ G‬ﻤﻥ ﺍﻟﺸﻜل‪:‬‬
‫*‬
‫⎧‬
‫⎫‬
‫⎬)‪F = ⎨ ρI q ; ρ ∈ ∆, ∆ = GF ( P‬‬
‫⎩‬
‫⎭‬
‫ﻋﻨﺩﺌ ٍﺫ ﺘﻭﺠﺩ ﻓﻲ ‪ G‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻨﻅﺎﻤﻴﺔ ‪ A‬ﻤﻌﺭﻓﺔ ﻜﻤﺎ ﻴﻠﻲ ]‪:[١‬‬
‫)‪A = a b F ; a q = b q = 1, (a, b) = ωI q ; ω q = 1,1 ≠ ω ∈ GF ( P‬‬
‫]‬
‫)‪(١‬‬
‫)‪(٢‬‬
‫[‬
‫⎤‪⎡ 0 1‬‬
‫⎢=‪b‬‬
‫‪, a = diag 1, ω, ω 2 ,..., ωq −1‬‬
‫⎥‬
‫⎦‪⎣I q −1 0‬‬
‫ﻟﺘﻜﻥ ‪ H‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻤﺎ ﻋﺎﺯﻟﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ ‪ G‬ﻭﺫﺍﺕ ﺭﺘﺒﺔ ﻓﺭﺩﻴﺔ ‪ r‬ﺒﺤﻴﺙ )‪r /| ( P − 1‬‬
‫ﻋﻨﺩﺌﺫ ﻤﻥ ﺃﺠل ﺘﺤﺩﻴﺩ ﺭﺘﺒﺔ ﺍﻟﻤﻨﻅﻡ ﻟﻬﺫﻩ ﺍﻟﺯﻤﺭﺓ ﺴﻨﻭﺭﺩ ﻤﺎ ﻴﻠﻲ‪:‬‬
‫ﻤﺒﺭﻫﻨﺔ‪:‬‬
‫ﺇﻥ ) ‪ N(H‬ﺭﺘﺒﺔ ﺍﻟﻤﻨﻅﻡ ﻷﻱ ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ‪ H‬ﻋﺎﺯﻟﺔ ﻏﻴﺭ ﺘﺒﺩﻴﻠﻴﺔ ﻤﻥ ﺍﻟﺯﻤﺭﺓ ‪ G‬ﻓـﻲ‬
‫ﺍﻟﺯﻤﺭﺓ )‪ GL(q,p‬ﻭﺫﺍﺕ ﺭﺘﺒﺔ ﻓﺭﺩﻴﺔ ‪ r‬ﺤﻴﺙ )‪ r /| (P − 1‬ﺘﺤﻘﻕ ﻤﺎ ﻴﻠﻲ‪:‬‬
‫‪ N(H) = G (١‬ﻋﻨﺩﻤﺎ ) ‪ ، p − 1 ≡/ 0(mod 4‬ﺃﻤ‪‬ﺎ ﺇﺫﺍ ﻜـﺎﻥ ) ‪P − 1 ≡ 0(mod 4‬‬
‫ﻓﻴﺠﺏ ﺃﻥ ﺘﻜﻭﻥ ‪.A⊃H‬‬
‫‪ N ( H ) = 6 q 2 ( p − 1) (٢‬ﻓﻴﻤﺎ ﻋﺩﺍ ﺫﻟﻙ‪.‬‬
‫ﺍﻟﺒﺭﻫﺎﻥ‪:‬‬
‫ﻟﺘﻜﻥ ‪ H‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻤﺎ ﻋﺎﺯﻟﺔ ﺫﺍﺕ ﺭﺘﺒﺔ ﻓﺭﺩﻴﺔ ‪ r‬ﻤﻥ ﺍﻟﺯﻤﺭﺓ ‪ ،G‬ﺤﻴﺙ )‪، r /| (P − 1‬‬
‫ﻋﻨﺩﺌﺫ ﻤﻥ ﺘﻌﺭﻴﻑ ﺍﻟﻌﺎﺯﻟﻴﺔ ﻓﺈﻥ )‪ N(H‬ﺯﻤﺭﺓ ﺍﻟﻤﻨﻅﻡ ﻟــ‪ H‬ﻓﻲ )‪ GL(q,p‬ﺘﺸـﻜل ﺯﻤـﺭﺓ‬
‫ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻓﻲ )‪ GL(q,p‬ﻭﺃﻴﻀﹰﺎ )‪ N(H‬ﺯﻤﺭﺓ ﺠﺯﺌﻴـﺔ ﻗﺎﺒﻠـﺔ ﻟﻠﺤـل ﻓـﻲ )‪،GL(q,p‬‬
‫ﺒﺎﻟﺤﻘﻴﻘﺔ‪ ،‬ﺇﻨﻪ ﻤﻥ ﺍﻟﺴﻬل ﺍﻟﺘﺤﻘﻕ ﺃﻥ ) ‪ H ⊇ N ′( H‬ﻋﻠﻤﹰﺎ ﺃﻥ ) ‪ N ′( H‬ﺍﻟﺯﻤـﺭﺓ ﺍﻟﻤﺸـﺘﻘﺔ‬
‫‪١٠٥‬‬
‫ﻴﻌﻘﻭﺏ ﻭ ﻋﻠﻲ ـ ﺩﺭﺍﺴﺔ ﻤﻨﻅﻡ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻭﺫﺍﺕ ﺍﻟﺘﺭﺒﺔ ﺍﻟﻔﺭﺩﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ …‬
‫ﻟﻠﺯﻤﺭﺓ )‪ N(H‬ﻫﻲ ﺃﺼﻐﺭ ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻨﻅﺎﻤﻴﺔ ﻓﻲ )‪ N(H‬ﺯﻤﺭﺘﻬـﺎ ﺍﻟﻌﺎﻤﻠـﺔ ﺘﺒﺩﻴﻠﻴـﺔ]‪،[٥‬‬
‫)‪N ( H‬‬
‫ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﺘﺒﺩﻴﻠﻴﺔ ﺫﻟﻙ ﻷﻥ‪:‬‬
‫ﻭﺒﺎﻟﺘﺎﻟﻲ ﺍﻟﺯﻤﺭﺓ ‪H‬‬
‫‪x, y ∈ N (H ) ⇒ xHyH = xyH = xy ( y −1 x −1 yx )H = yxH = yHxH‬‬
‫ﺒﻤﺎ ﺃﻥ ﻜل ﺯﻤﺭﺓ ﺘﺒﺩﻴﻠﻴﺔ ﻫﻲ ﺯﻤﺭﺓ ﻗﺎﺒﻠﺔ ﻟﻠﺤل‪ ،‬ﻭﺒﻤﺎ ﺃﻥ ﺍﻟﺯﻤﺭ ﺍﻟﻤﻌﻁﻴﺔ ‪ H‬ﺃﻴﻀﹰﺎ ﺯﻤﺭﺓ‬
‫ﺠﺯﺌﻴﺔ ﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻋﻨﺩﺌﺫ ﺒﺤﺴﺏ ﺨﻭﺍﺹ ﺍﻟﺯﻤﺭ ﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل]‪ ،[٥‬ﺘﻜﻭﻥ )‪ N(H‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ‬
‫ﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ ،GL(q,p‬ﻭﺒﺎﻟﺘﺎﻟﻲ ﻤﻥ ﺃﻋﻅﻤﻴﺔ ‪ G‬ﻓﻲ )‪ GL(q,p‬ﻓﺈﻥ‪:‬‬
‫‪N(H) ⊆ G‬‬
‫)‪(٣‬‬
‫ﺍﻵﻥ‪ ،‬ﺒﺤﺴﺏ ]‪ [٢‬ﺇﺫﺍ ﻜﺎﻨﺕ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﺍﻵﻨﻔﺔ ﺍﻟﺫﻜﺭ ﻏﻴﺭ ﺘﺒﺩﻴﻠﻴﺔ ﻓﻲ‪ G‬ﻓـﺈﻥ‬
‫‪ H‬ﺘﺤﻭﻱ ﺃﺼﻐﺭ ﺯﻤﺭﺓ ﻨﻅﺎﻤﻴﺔ ﻋﺎﺯﻟﺔ ﻭﻏﻴﺭ ﺘﺒﺩﻴﻠﻴﺔ ‪ B‬ﻓﻲ ‪ G‬ﻭﻟﻬﺎ ﺍﻟﺸﻜل‪:‬‬
‫; ‪B= a b ω‬‬
‫‪ aω,b,‬ﻤﻌﺭﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪.(٢‬‬
‫ﻥ‪:‬‬
‫ﻭﺒﺎﻟﺘﺎﻟﻲ ﻤﻥ ﺍﻟﻭﺍﻀﺢ ﺠﺩﹰﺍ ﺃ ‪‬‬
‫)‪(٤‬‬
‫)‪A⊆N(H‬‬
‫ﺤﻴﺙ ‪ A‬ﻤﻌﺭﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪:(٢‬‬
‫ﺃﻴﻀﹰﺎ ﺒﺤﺴﺏ ]‪ [٢‬ﺘﻭﺠﺩ ﻤﻥ ﺃﺠـل ﺃﻱ ﺠﺯﺌﻴـﺔ ﻋﻅﻤـﻰ ﻋﺎﺯﻟـﺔ ﻗﺎﺒﻠـﺔ ﻟﻠﺤـل ﻓـﻲ‬
‫)‪ ،G L(q,p‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ‪ U‬ﻋﻅﻤﻰ ﻋﺎﺯﻟﺔ ﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ ،SL(٢,q‬ﺒﺤﻴﺙ ﻴﻜﻭﻥ ﺍﻟﺘﻁﺒﻴﻕ‬
‫ﻼ ﺯﻤﺭﻴﹰﺎ ﻏﺎﻤﺭﹰﺍ‪:‬‬
‫‪ Ψ‬ﺍﻟﻤﻌﺭ‪‬ﻑ ﻜﻤﺎ ﻴﻠﻲ‪ ،‬ﺘﺸﺎﻜ ﹰ‬
‫⎤ ‪⎡α β‬‬
‫⎢ ‪Ψ : N(A) → U; x a‬‬
‫⎥‬
‫⎦‪⎣ γ δ‬‬
‫ﺤﻴﺙ )‪ α, β, δ, γ ∈ GF(q‬ﻭﺘﺭﺘﺒﻁ ﻓﻴﻤﺎ ﺒﻴﻨﻬﺎ ﺒﺎﻟﻌﻼﻗﺎﺕ‪:‬‬
‫)‪(٥‬‬
‫‪xax −1 = λa α bγ , xbx −1 = µα β bδ‬‬
‫ﺤﻴﺙ‪ ∈a , b A :‬ﻭ ‪ ∆ → µ ،λ‬ﻭ ‪ A‬ﻭ ∆ ﻤﻌﺭﻓﺎﻥ ﺒﺎﻟﻌﻼﻗﺔ )‪ (٢‬ﻭﺍﻟﻌﻼﻗﺔ )‪ (١‬ﻋﻠﻰ ﺍﻟﺘﺭﺘﻴﺏ‪:‬‬
‫ﺇﺫﹰﺍ‪:‬‬
‫‪≅U‬‬
‫)‪N ( A‬‬
‫)‪(٦‬‬
‫‪A‬‬
‫‪١٠٦‬‬
‫ﻤﺠﻠﺔ ﺠﺎﻤﻌﺔ ﺩﻤﺸﻕ ﻟﻠﻌﻠﻭﻡ ﺍﻷﺴﺎﺴﻴﺔ ـ ﺍﻟﻤﺠﻠﺩ )‪ (١٦‬ـ ﺍﻟﻌﺩﺩ ﺍﻷﻭل ـ ‪٢٠٠٠‬‬
‫ﻵﻥ ‪ ، KerΨ=A‬ﺤﻴﺙ ‪ A‬ﻤﻌﺭﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪ ،(٢‬ﻭﻤﻥ ﺨﻭﺍﺹ ﺍﻟﺯﻤـﺭ ﺍﻟﻘﺎﺒﻠـﺔ ﻟﻠﺤـل‬
‫ﻥ )‪ N(A‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ ،GL(q,p‬ﻭﺃﻴﻀـﹰﺎ‬
‫ﻭﺒﺎﻻﻋﺘﻤﺎﺩ ﻋﻠﻰ ﺍﻟﻌﻼﻗﺔ )‪ ،(٦‬ﻓﺈ ‪‬‬
‫ﻋﺎﺯﻟﺔ ﻻﺤﺘﻭﺍﺌﻬﺎ ﺍﻟﺯﻤﺭﺓ ﺍﻟﻌﺎﺯﻟﺔ ‪ ،A‬ﻭﻤﻥ ﺃﻋﻅﻤﻴﺔ ‪ G‬ﻓﻲ )‪ GL(q,p‬ﻓﺈﻥ )‪ ،G⊇N(A‬ﻭﺒﻤـﺎ‬
‫ﻥ ‪ G∆A‬ﺇﺫﹰﺍ‪:‬‬
‫ﺃ‪‬‬
‫)‪(٧‬‬
‫‪N(A)=G‬‬
‫ﺍﻵﻥ‪ ،‬ﺒﺤﺴﺏ ]‪ [٢‬ﺘﻭﺠﺩ ﻤﻥ ﺃﺠل ﺃﻱ ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ‪ ،H‬ﻋﺎﺯﻟﺔ ﻭﻏﻴﺭ ﺘﺒﺩﻴﻠﻴـﺔ ﻤـﻥ ‪،G‬‬
‫ﻭﺫﺍﺕ ﺭﺘﺒﺔ ﻓﺭﺩﻴﺔ ‪ r‬ﺤﻴﺙ )‪ r /| ( P − 1‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ‪ u‬ﺩﻭﺭﻴﺔ ﻭﺫﺍﺕ ﺭﺘﺒـﺔ ﻓﺭﺩﻴـﺔ ﻤـﻥ‬
‫ﺍﻟﺯﻤﺭﺓ ‪ U‬ﺍﻟﻤﻌﺭ‪‬ﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪ (٥‬ﺒﺤﻴﺙ‪:‬‬
‫‪(I‬‬
‫‪(II‬‬
‫ﻥ ‪ ، u = I‬ﺤﻴﺙ ‪ I‬ﻋﻨﺼﺭ ﺍﻟﻭﺤﺩﺓ ﻓﻲ ‪.U‬‬
‫ﻋﻨﺩﻤﺎ ﺘﻜﻭﻥ ‪ A⊃H‬ﻓﺈ ‪‬‬
‫ـﺔ ﻭ ‪، u = τ‬‬
‫ﻥ ‪ u‬ﻋﺎﺯﻟـــ‬
‫⊃ ‪ A‬ﻓـــﺈ ‪‬‬
‫ـﺩﻤﺎ ﺘﻜـــ‬
‫ﻭﻋﻨـــ‬
‫ـﻭﻥ ‪/ H‬‬
‫⎤‪⎡1 q − 3‬‬
‫⎢ = ‪ ، U ∋ τ ، τ 3 = I ، τ‬ﻭﻓﻲ ﻫﺫﻩ ﺍﻟﺤﺎﻟﺔ ﻴﻜﻭﻥ ﻟﺩﻴﻨﺎ ﺃﻴﻀﹰﺎ ﻤﺎ ﻴﻠﻲ‪:‬‬
‫ﺤﻴﺙ ⎥‬
‫⎦‪⎣1 q − 2‬‬
‫‪ U∆u (١‬ﻋﻨﺩﻤﺎ )‪. p − 1 ≡/ 0(mod 4‬‬
‫َ‬
‫َ‬
‫‪ ، U ⊃ u δ ∆u (٢‬ﺤﻴﺙ ) ‪ ، uδ = N (u‬ﻭ ‪ uδ = δ‬ﺤﻴﺙ‬
‫⎤‪⎡ − 1 q − 2‬‬
‫⎢ = ‪ ، U ∋ δ , δ 6 = I , δ‬ﻭﻫـــﺫﻩ ﺍﻟﺤﺎﻟـــﺔ ﻤﺤﻘﻘـــﺔ ﻋﻨـــﺩﻤﺎ‬
‫⎥‬
‫⎦‪⎣ − 1 q − 3‬‬
‫)‪. p − 1 ≡ 0(mod 4‬‬
‫ﻤﻥ ﺘﻌﺭﻴﻑ ﺍﻟﻌﺎﺯﻟﻴﺔ ﻓﺈﻥ )‪ N(u‬ﺯﻤﺭﺓ ﺍﻟﻤﻨﻅﻡ ﻟــ‪ u‬ﻓﻲ )‪ ،SL(٢,q‬ﺘﺸﻜل ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ‬
‫ﻋﺎﺯﻟﺔ ﻻﺤﺘﻭﺍﺌﻬﺎ ﻓﻲ ﺍﻟﺤﺎﻟﺔ )‪ (I‬ﺍﻟﺯﻤﺭﺓ ﺍﻟﻌﻅﻤﻰ ﺍﻟﻌﺎﺯﻟﺔ ﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ‪ ،U‬ﻭﻻﺤﺘﻭﺍﺌﻬﺎ ﺍﻟﻌﻨﺼﺭ‬
‫ﺍﻟﻌﺎﺯل ‪ τ‬ﻓﻲ ﺍﻟﺤﺎﻟﺔ )‪ ،(II‬ﻜﻤﺎ ﺃﻨﻪ ﺃﻴﻀﹰﺎ ﻴﻤﻜﻥ ﺍﻟﺘﺤﻘﻕ ﻤﻥ ﺃﻥ ﺍﻟﺯﻤﺭﺓ )‪ N(u‬ﺘﺸـﻜل ﺯﻤـﺭﺓ‬
‫ﺠﺯﺌﻴﺔ ﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ SL(٢,q‬ﺒﺎﻋﺘﻤﺎﺩ ﺍﻷﺴﻠﻭﺏ ﻨﻔﺴﻪ ﺍﻟﻤﺘﺒﻊ ﻓﻲ ﺍﻟﺒﺭﻫﺎﻥ ﻋﻠﻰ ﻗﺎﺒﻠﻴﺔ ﺍﻟﺤل‬
‫ﻟﺯﻤﺭﺓ ﺍﻟﻤﻨﻅﻡ ﻟﻠﺯﻤﺭﺓ ﺍﻟﻌﺎﺯﻟﺔ‪ ،‬ﻭﺍﻟﻭﺍﺭﺩ ﻓﻲ ﺍﻟﻤﺒﺭﻫﻨﺔ ﺍﻟﺴﺎﺒﻘﺔ‪ ،‬ﻭﻤﻥ ﺃﻋﻅﻤﻴﺔ ‪ U‬ﻓﻲ )‪SL(٢,q‬‬
‫ﻓﺈﻥ‪:‬‬
‫)‪(٨‬‬
‫‪N(u)⊆U‬‬
‫‪١٠٧‬‬
‫ﻴﻌﻘﻭﺏ ﻭ ﻋﻠﻲ ـ ﺩﺭﺍﺴﺔ ﻤﻨﻅﻡ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻭﺫﺍﺕ ﺍﻟﺘﺭﺒﺔ ﺍﻟﻔﺭﺩﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ …‬
‫ﻤﻥ ﻨﺎﺤﻴﺔ ﺜﺎﻨﻴﺔ‪ :‬ﺒﻤﺎ ﺃﻨﹼﻪ ﻴﻭﺠﺩ ﺘﻘﺎﺒل ‪ ١‬ﻟـ‪ ١‬ﺒﻴﻥ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﻓﻲ ﺍﻟﻤﻨﻁﻠـﻕ ﻭﺍﻟﺘـﻲ‬
‫ﻥ ﺯﻤـﺭﺓ‬
‫ﺘﺤﻭﻱ ﺍﻟﻨﻭﺍﺓ ‪ A‬ﻭﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﻓﻲ ﺍﻟﻤﺴﺘﻘﺭ‪ ،‬ﻭﺒﺎﻻﻋﺘﻤﺎﺩ ﻋﻠﻰ ﺍﻟﻌﻼﻗـﺔ )‪ (٤‬ﻓـﺈ ‪‬‬
‫ﺍﻟﻤﻨﻅﻡ ﻟـﹻ‪ H‬ﻓﻲ )‪ GL(q,p‬ﺘﺭﺘﺒﻁ ﻤﻊ ﺯﻤﺭﺓ ﺍﻟﻤﻨﻅﻡ ﻟـﹻ‪ u‬ﻓﻲ )‪ SL(٢,q‬ﺒﺎﻟﻌﻼﻗﺔ ﺍﻟﺘﺎﻟﻴﺔ‪:‬‬
‫) ‪Ψ1 ( N (H )) = N (u ); Ψ1 = Ψ | N ( H‬‬
‫)‪(٩‬‬
‫ﻭﻫﺫﻩ ﺍﻟﻌﻼﻗﺔ ﻤﺤﻘﻘﺔ ﺩﻭﻤﹰﺎ ﺒﺤﺴﺏ ﺍﻟﺒﻨﺎﺀ ﻟﻠﺘﺸﺎﻜل ﺍﻟﺯﻤﺭﻱ ‪ Ψ‬ﺍﻟﻤﻌﺭﻑ ﺒﺎﻟﻌﻼﻗﺔ )‪ (٥‬ﻤـﻊ‬
‫ﻼ ﺯﻤﺭﻴﹰﺎ ﻏﺎﻤﺭﺍﹰ‪،‬‬
‫ﺍﻷﺨﺫ ﺒﺎﻟﺤﺴﺒﺎﻥ ﺘﻌﺭﻴﻑ ﺍﻟﻤﻨﻅﻡ ﻟﻠﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴﺔ‪ ،‬ﻜﻤﺎ ﺃﻥ ‪ Ψ1‬ﻴﺸﻜل ﺘﺸﺎﻜ ﹰ‬
‫ﻨﻭﺍﺘﻪ ‪ ، ker Ψ1 = A‬ﺤﻴﺙ ‪ A‬ﻤﻌﺭﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪ ،(٢‬ﻭﺒﺎﻟﺘﺎﻟﻲ‪:‬‬
‫) ‪N (H ) ≅ N (u‬‬
‫)‪(١٠‬‬
‫‪A‬‬
‫ﻥ‬
‫‪ (١‬ﻤﻥ )‪ (II‬ﻟﺩﻴﻨﺎ ‪ ، U > u‬ﻭﻤـﻥ ﺍﻟﻌﻼﻗـﺔ )‪ (٨‬ﻴﻨـﺘﺞ ﺃ ‪‬‬
‫ﺍﻵﻥ‪ ،‬ﺒﺤﺴﺏ )‪ (I‬ﻭﺒﺤﺴﺏ ) َ‬
‫‪ ،N(u)=U‬ﻨﺤﺼل ﺃﻴﻀﹰﺎ ﻤﻥ ﺍﻟﻌﻼﻗﺘﻴﻥ )‪ (٦‬ﻭ )‪ (١٠‬ﻋﻠﻰ‪:‬‬
‫)‪N ( H ) = A .U = N ( A‬‬
‫ﻥ ‪. N (H ) = G‬‬
‫ﻭﻤﻥ ﺍﻟﻌﻼﻗﺔ )‪ (٧‬ﻓﺈ ‪‬‬
‫ﻥ‪:‬‬
‫‪ (٢‬ﻤﻥ )‪ (II‬ﻭﺒﺎﻻﻋﺘﻤﺎﺩ ﻋﻠﻰ ﺍﻟﻌﻼﻗﺔ )‪ (١٠‬ﻴﻨﺘﺞ ﺃ ‪‬‬
‫ﻭﺒﺤﺴﺏ ﺍﻟﺤﺎﻟﺔ ) َ‬
‫‪N (H ) = A . uδ‬‬
‫ﻥ‪:‬‬
‫ﻭﻟﻜﻥ‪ ،‬ﺒﺤﺴﺏ ﺍﻟﺒﻨﺎﺀ ﻟـﹻ‪ A‬ﺍﻟﻤﻌﺭ‪‬ﻑ ﺒﺎﻟﻌﻼﻗﺔ )‪ (٢‬ﻓﺈ ‪‬‬
‫)‪A = q 2 ( p − 1‬‬
‫ﻭﻤﻨﻪ‪:‬‬
‫)‪N (H ) = 6 q 2 ( p − 1‬‬
‫* ﺤﺎﻟﺔ ﺨﺎﺼﺔ‪:‬‬
‫ﻤﻥ ﺃﺠل ﺃﻱ ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ‪ H‬ﻋﺎﺯﻟﺔ ﺘﺒﺩﻴﻠﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ ‪ G‬ﻭﺫﺍﺕ ﺭﺘﺒﺔ ﻓﺭﺩﻴﺔ ‪ ،r‬ﺤﻴـﺙ‬
‫ﻥ‪:‬‬
‫)‪ r /| ( P − 1‬ﻓﺈ ‪‬‬
‫‬‫‪-‬‬
‫)‪ N (H ) = (q − 1)q 2 ( p − 1‬ﻋﻨﺩﻤﺎ ‪. A ⊃ H‬‬
‫)‪ N (H ) = 3 ( p − 1‬ﻓﻴﻤﺎ ﻋﺩﺍ ﺫﻟﻙ‪.‬‬
‫‪١٠٨‬‬
‫ﻤﺠﻠﺔ ﺠﺎﻤﻌﺔ ﺩﻤﺸﻕ ﻟﻠﻌﻠﻭﻡ ﺍﻷﺴﺎﺴﻴﺔ ـ ﺍﻟﻤﺠﻠﺩ )‪ (١٦‬ـ ﺍﻟﻌﺩﺩ ﺍﻷﻭل ـ ‪٢٠٠٠‬‬
‫ﺍﻟﺒﺭﻫﺎﻥ‪:‬‬
‫ﻟﺘﻜﻥ ‪ H‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻤﺎ ﻋﺎﺯﻟﺔ ﺘﺒﺩﻴﻠﻴﺔ‪ ،‬ﻭﺫﺍﺕ ﺭﺘﺒﺔ ﻓﺭﺩﻴﺔ ‪ r‬ﺤﻴﺙ )‪ ، r /| ( P − 1‬ﻋﻨﺩﺌﺫ‬
‫ﻭﺒﺤﺴﺏ ﺍﻟﻤﺒﺭﻫﻨﺔ ﻓﺈ ‪‬‬
‫ﻥ ) ‪ ، G ⊇ N (H‬ﺍﻵﻥ‪ ،‬ﻤﻥ ﺃﺠل ﺘﺤﺩﻴﺩ ﺭﺘﺒﺔ ﺍﻟﻤﻨﻅﻡ ﻟـﹻ‪ H‬ﺴﻨﺩﺭﺱ‬
‫ﺍﻟﺤﺎﻟﺘﻴﻥ ﺍﻟﺘﺎﻟﻴﺘﻴﻥ‪:‬‬
‫ﻥ ﺍﻟﺸﻜل ﺍﻟﻌﺎﻡ ﻟـﹻ‪ H‬ﺴﻴﻜﻭﻥ ﻜﻤﺎ ﻴﻠﻲ‪:‬‬
‫‪ ( I ′‬ﺇﺫﺍ ﻜﺎﻨﺕ ‪ A ⊃ H‬ﻓﺈ ‪‬‬
‫‪H = m F1 ; m = a i b j ; m = q,0 ≤ i ≤ q − 1,0〈 j ≤ q − 1‬‬
‫)‪(١١‬‬
‫ﺤﻴﺙ ‪ a‬ﻭ ‪ ، A ∋ b‬ﻭ ‪ F1‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻤﻥ ﺍﻟﺯﻤﺭﺓ ‪ ،F‬ﺘﺤﻭﻱ ﺍﻟﻌﻨﺼﺭ ‪ ، ω I q‬ﺤﻴﺙ‬
‫‪ ω‬ﻭ ‪ F‬ﻭ ‪ A‬ﻤﻌﺭ‪‬ﻓﺔ ﺒﺎﻟﻌﻼﻗﺘﻴﻥ )‪ (١‬ﻭ )‪:(٢‬‬
‫ﻥ ﺠﻤﻴﻊ ﺍﻟﺯﻤﺭ ﺍﻟﻌﺎﺯﻟﺔ ﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻤﻥ ﺍﻟﺭﺘﺒﺔ ‪ r‬ﺍﻟﻤﺤﻘﻘﺔ ﻟﻠﺸﺭﻭﻁ‬
‫ﻤﻥ ﻨﺎﺤﻴﺔ ﺜﺎﻨﻴﺔ‪ ،‬ﺒﻤﺎ ﺃ ‪‬‬
‫ﺍﻟﺴﺎﺒﻘﺔ‪ ،‬ﺘﺸﻜل ﺼﻔﹰﺎ ﻤﻥ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻤﺘﺭﺍﻓﻘﺔ ﻓﻲ )‪ GL(q,p‬ﻓﺈﻨﹼﻪ ﻴﻜﻔﻲ ﻟﺘﺤﺩﻴـﺩ ﺭﺘﺒـﺔ‬
‫ﺍﻟﻤﻨﻅﻡ ﻟﻠﺯﻤﺭ ﻤﻥ ﻫﺫﺍ ﺍﻟﺼﻑ‪ ،‬ﺃﻥ ﺘﺤﺩﺩ ﺭﺘﺒﺔ ﺍﻟﻤﻨﻅﻡ ﻟﻤﻤﺜل ﻭﺍﺤﺩ ﻤﻥ ﻫﺫﺍ ﺍﻟﺼﻑ ﻤﻥ ﻭﺠﻬﺔ‬
‫ﻨﻅﺭ ﺍﻟﺘﺭﺍﻓﻕ‪ ،‬ﻭﻟﻴﻜﻥ ﻫﺫﺍ ﺍﻟﻤﻤﺜل ﻫﻭ‪:‬‬
‫‪H = b F1 ; b q = 1‬‬
‫ﻭ ‪ b‬ﻤﻌﺭ‪‬ﻑ ﺒﺎﻟﻌﻼﻗﺔ )‪ ،(٢‬ﻭ ‪ F1‬ﻤﻌﺭ‪‬ﻑ ﺒﺎﻟﻌﻼﻗﺔ )‪.(١١‬‬
‫ﻤﻥ ﺃﺠل ﺫﻟﻙ ﺘﻭﺠﺩ ﻓﻲ ﺍﻟﺯﻤﺭﺓ ‪ U‬ﺍﻟﻤﻌﺭ‪‬ﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪ (٥‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ‪ um‬ﻤﺜﻠﺜﻴـﺔ ﻤـﻥ‬
‫ﺍﻷﺴﻔل‪ ،‬ﻭﻟﻬﺎ ﺍﻟﺭﺘﺒﺔ ‪ ،q-١‬ﺩﻭﺭﻴﺔ‬
‫‪ u m = f‬ﻤﻭ‪‬ﻟﺩﺓ ﺒﺎﻟﻌﻨﺼﺭ ‪ f‬ﺒﺤﻴﺙ‪:‬‬
‫⎤ ‪⎡α 0‬‬
‫⎢= ‪f‬‬
‫) ‪⎥; α, γ, δ ∈ GF(q‬‬
‫⎦⎥‪⎣⎢ γ δ‬‬
‫ﻭﺍﻟﺯﻤﺭﺓ ‪ um‬ﻤﻭﺠـﻭﺩﺓ ﺩﻭﻤـﹰﺎ ﺒﺤﺴـﺏ ﺍﻟﺒﻨـﺎﺀ ﻟـــﹻ‪ U‬ﻜﻤـﺎ ﻓـﻲ ]‪ ،[٢‬ﻋﻨﺩﺌـﺫ‬
‫ﻼ ﺯﻤﺭﻴﹰﺎ ﻏﺎﻤﺭﹰﺍ‪:‬‬
‫ﺍﻟﺘﻁﺒﻴﻕ ) ‪ Ψm = Ψ | N ( H‬ﺍﻟﻤﻌﺭ‪‬ﻑ ﻜﻤﺎ ﻴﻠﻲ ﻴﺸﻜل ﺘﺸﺎﻜ ﹰ‬
‫⎤ ‪⎡α 0‬‬
‫⎢ ‪Ψm : N (H ) ⊆ G → u m ; x a‬‬
‫)‪⎥ (١٢‬‬
‫⎦ ‪⎣γ δ‬‬
‫ﺤﻴﺙ ‪ GF (q ) ∋ δ , γ , α‬ﻭﺘﺭﺘﺒﻁ ﻓﻴﻤﺎ ﺒﻴﻨﻬﺎ ﺒﺎﻟﻌﻼﻗﺎﺕ‪:‬‬
‫‪xax −1 = λa α b γ , xbx −1 = b δ‬‬
‫‪١٠٩‬‬
‫ﻴﻌﻘﻭﺏ ﻭ ﻋﻠﻲ ـ ﺩﺭﺍﺴﺔ ﻤﻨﻅﻡ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻭﺫﺍﺕ ﺍﻟﺘﺭﺒﺔ ﺍﻟﻔﺭﺩﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ …‬
‫ﺤﻴﺙ ‪ a‬ﻭ ‪ ، A ∋ b‬ﻭ ‪ A, ∆ ∋ λ‬ﻭ ∆ ﻤﻌﺭﻓﺔ ﺒﺎﻟﻌﻼﻗﺘﻴﻥ )‪ (٢‬ﻭ )‪.(١‬‬
‫ﻼ ﺯﻤﺭﻴﺎﹰ‪ ،‬ﻤﻥ ﺃﺠل ﺫﻟﻙ ﻴﻜﻔـﻲ‬
‫ﻥ ﺍﻟﺘﻁﺒﻴﻕ ‪ Ψm‬ﻴﺸﻜل ﺘﺸﺎﻜ ﹰ‬
‫ﻤﻥ ﺍﻟﻤﻤﻜﻥ ﺍﻟﺒﺭﻫﺎﻥ ﻋﻠﻰ ﺃ ‪‬‬
‫ﺍﻟﺘﺤﻘﻕ ﺃﻥ ﺼﻭﺭﺓ ﺠﺩﺍﺀ ﺃﻱ ﻋﻨﺼﺭﻴﻥ ﻤﻥ ﺍﻟﺯﻤﺭﺓ ﺍﻟﺠﺯﺌﻴﺔ )‪ N(H‬ﺘﺴﺎﻭﻱ ﺠﺩﺍﺀ ﺼـﻭﺭﻫﻤﺎ‬
‫ﺃﻱ ﻤﻬﻤﺎ ﻴﻜﻥ ‪ N(H)∋x,y‬ﻓﺈﻥ ﺍﻟﻤﺴﺎﻭﺍﺓ ﺍﻟﺘﺎﻟﻴﺔ ﻴﺠﺏ ﺃﻥ ﺘﻜﻭﻥ ﺼﺤﻴﺤﺔ‪:‬‬
‫) ‪Ψm ( xy) = Ψm ( x)Ψm ( y‬‬
‫)*(‬
‫⎤ ‪⎡α 1 0‬‬
‫⎢ = )‪Ψm ( x‬‬
‫ﻨﻔﺭﺽ ﺃﻥ ⎥‬
‫⎦ ‪⎣γ 1 δ 1‬‬
‫ﻋﻨﺩﺌﺫ ﻓﺈﻥ‪:‬‬
‫‪xbx −1 = b δ 1‬‬
‫‪æ‬‬
‫⎤ ‪⎡α 2 0‬‬
‫⎢ = )‪Ψm ( y‬‬
‫⎥‬
‫⎦ ‪⎣γ 2 δ 2‬‬
‫‪xax −1 = λ 1a α 1 b γ 1 ,‬‬
‫ﺃﻴﻀﹰﺎ‪:‬‬
‫‪yby −1 = bδ 2‬‬
‫‪yay −1 = λ 2 a α 2 b γ 2 ,‬‬
‫ﺤﻴﺙ ‪i = 1,2 æ GF (q) ∋ α i , γ i , δ i æ ∆ ∋ λ i‬‬
‫ﻭﺒﺤﺴﺏ ﺨﻭﺍﺹ ﺍﻟﻌﻨﺎﺼﺭ ﺍﻟﻤﺒﺎﺩﻟﺔ‪ ،‬ﻭﺒﻌﺩ ﺴﻠﺴﻠﺔ ﻤﻥ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻟﺠﺒﺭﻴﺔ‪ ،‬ﻴﻤﻜﻥ ﺍﻟﺘﺤﻘـﻕ‬
‫ﻥ‪:‬‬
‫ﺃ‪‬‬
‫‪xyby −1 x −1 = bδ 1δ 2‬‬
‫‪xyay −1 x −1 = µa α 1α 2 b γ 1α 2 +δ 1γ 2 ,‬‬
‫ﺤﻴﺙ ‪∆ = GF ( P) ∋ µ‬‬
‫ﻭﺒﺎﻟﺘﺎﻟﻲ ﺒﺤﺴﺏ ﺍﻟﺒﻨﺎﺀ ﻟﻠﺩﺍﻟﺔ ‪ Ψm‬ﺍﻟﻤﻌﺭ‪‬ﻑ ﺒﺎﻟﻌﻼﻗﺔ )‪ (١٢‬ﻓﺈﻥ ﺍﻟﻤﺴﺎﻭﺍﺓ ﺘﻜـﻭﻥ ﻤﺤﻘﻘـﺔ‬
‫ﺒﻴﻥ ﻁﺭﻓﻲ ﺍﻟﻌﻼﻗﺔ )*( ﻭﺒﺎﻟﺘﺎﻟﻲ ﻓﺈﻥ ‪ Ψm‬ﺘﺸﺎﻜل ﺯﻤﺭﻱ ﻭﺃﻴﻀﹰﺎ ﻏـﺎﻤﺭ‪ ،‬ﺒﺎﻟﺤﻘﻴﻘـﺔ ﻟـﻴﻜﻥ‬
‫‪ ، SL(2, q ) ⊃ u m ∋ f‬ﻭﻟﻴﻜﻥ‬
‫‪ A ∋ c, d‬ﺒﺤﻴـﺙ ‪ c = λa α b γ‬ﻭ ‪ ، d = b δ‬ﻭﺒﺤﻴـﺙ‬
‫ﺘﻜﻭﻥ ﺍﻟﻘﻭﻯ ﺍﻟﻤﺄﺨﻭﺫﺓ ‪ GF (q ) ∋ δ , γ , α‬ﺒﻤﺜﺎﺒﺔ ﻋﻨﺎﺼﺭ ﻟﻠﻤﺼﻔﻭﻓﺔ ‪ ،h‬ﻜﺫﻟﻙ ‪A ∋ b ،a‬‬
‫ﻭ ‪ ، ∆ ∋ λ‬ﺤﻴﺙ ‪ A‬ﻭ ∆ ﻤﻌﺭ‪‬ﻓﺘﺎﻥ ﺒﺎﻟﻌﻼﻗﺘﻴﻥ )‪ (٢‬ﻭ )‪.(١‬‬
‫ﻥ‪:‬‬
‫ﺒﻔﺭﺽ ‪ λ = 1‬ﻭﺒﺎﻻﻋﺘﻤﺎﺩ ﻋﻠﻰ ﺍﻟﻌﻼﻗﺔ )‪ ،(٢‬ﻨﺠﺩ ﺃ ‪‬‬
‫‪= ω αδ‬‬
‫‪(c, d ) = (a α bγ , bδ ) = (a α , bδ )(bγ , bδ ) = (a α , bδ ) = (a, b)αδ‬‬
‫‪١١٠‬‬
‫ﻤﺠﻠﺔ ﺠﺎﻤﻌﺔ ﺩﻤﺸﻕ ﻟﻠﻌﻠﻭﻡ ﺍﻷﺴﺎﺴﻴﺔ ـ ﺍﻟﻤﺠﻠﺩ )‪ (١٦‬ـ ﺍﻟﻌﺩﺩ ﺍﻷﻭل ـ ‪٢٠٠٠‬‬
‫⎤ ‪⎡α 0‬‬
‫ﻭﺒﻤﺎ ﺃ ‪‬‬
‫⎢ = ‪ SL(2, q ) ∋ h‬ﺇﺫﺍ ‪.det h = αδ=١‬‬
‫ﻥ ⎥‬
‫⎦ ‪⎣γ δ‬‬
‫ﻭﻤﻨﻪ‪:‬‬
‫) ‪(c, d ) = ω αδ = ω = (a, b‬‬
‫ﺃﻴﻀﹰﺎ‪:‬‬
‫‪b q = d q = 1, a q = c q = 1‬‬
‫ﻋﻨﺩﺌﺫ ﺒﺤﺴﺏ ]‪ [١‬ﻓﺈﻨﹼﻪ ﻴﻭﺠﺩ ﻋﻨﺼﺭ ‪ GL(q, p ) ∋ x‬ﺒﺤﻴﺙ‪:‬‬
‫)‪(1′′‬‬
‫)‪(١٣‬‬
‫)‪(2′′‬‬
‫‪xax −1 = c = λa α b γ‬‬
‫‪δ‬‬
‫‪=d =b‬‬
‫‪−1‬‬
‫‪xbx‬‬
‫ﻥ ‪ ، N (H ) ∋ x‬ﻜـﺫﻟﻙ‬
‫‪ (٢‬ﻓﺈ ‪‬‬
‫ﻥ ‪ ، G = N ( A) ∋ x‬ﻭﺃﻴﻀﹰﺎ ﻤﻥ ﺍﻟﻌﻼﻗﺔ ) ً‬
‫ﻤﻥ ﺍﻟﻭﺍﻀﺢ ﺃ ‪‬‬
‫ﻤﻥ ﺍﻟﻌﻼﻗﺘﻴﻥ )‪ (١٢‬ﻭ )‪ (١٣‬ﻓﺈ ‪‬‬
‫ﻥ ‪ ، Ψm ( x ) = h‬ﻭﺒﺎﻟﺘﺎﻟﻲ ‪ Ψm‬ﻏﺎﻤﺭ‪.‬‬
‫ﻥ ‪ ، ker Ψm = A‬ﺤﻴﺙ ‪ A‬ﻤﻌﺭ‪‬ﻓـﺔ‬
‫ﺍﻵﻥ‪ ،‬ﺒﺎﻻﻋﺘﻤﺎﺩ ﻋﻠﻰ ﺍﻟﻌﻼﻗﺘﻴﻥ )‪ (٢‬ﻭ )‪ (١٢‬ﻨﺠﺩ ﺃ ‪‬‬
‫ﺒﺎﻟﻌﻼﻗﺔ )‪ ،(٢‬ﻭﺒﺎﻟﺘﺎﻟﻲ ‪ N (H ) ≅ u m‬ﻭﻤﻨﻪ‪:‬‬
‫‪A‬‬
‫‪2‬‬
‫)‪N (H ) = A . u m = (q − 1)q ( p − 1‬‬
‫⊃ ‪ A‬ﻓﺈﻨﹼﻪ ﺒﺤﺴﺏ ]‪ [٢‬ﺴﻴﻜﻭﻥ ﻟﻬﺎ ﺍﻟﺸﻜل ﺍﻟﺘﺎﻟﻲ‪:‬‬
‫‪ ( II ′‬ﺃﻤ‪‬ﺎ ﺇﺫﺍ ﻜﺎﻨﺕ ‪/ H‬‬
‫)‪H = x i F2 ; x = 3q; i = 1, q (١٤‬‬
‫ﺤﻴﺙ ‪ F٢‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻤﻥ ﺍﻟﺯﻤﺭﺓ ‪ ،F‬ﻜﺫﻟﻙ ‪ ، x ∩ F = ω‬ﺤﻴﺙ ‪ ω‬ﻭ ‪ F‬ﻤﻌﺭ‪‬ﻓﺎﻥ‬
‫ﺒﺎﻟﻌﻼﻗﺘﻴﻥ )‪ (٢‬ﻭ )‪ ،(١‬ﻭ ‪Ψ ( x ) = τ‬‬
‫ﺤﻴﺙ ‪ τ 3 = I‬ﻭ ‪ U ∋ τ ≠ I‬ﻭﻫـﻭ ﻤﻌـﺭ‪‬ﻑ‬
‫ﺒﺎﻟﺤﺎﻟﺔ )‪ (II‬ﺍﻟﻭﺍﺭﺩﺓ ﻓﻲ ﺍﻟﻤﺒﺭﻫﻨﺔ ﺍﻟﺴﺎﺒﻘﺔ ﻭﺒﺤﻴﺙ ‪ Ψ‬ﻭ ‪ U‬ﻤﻌﺭ‪‬ﻓﺎﻥ ﺒﺎﻟﻌﻼﻗﺔ )‪.(٥‬‬
‫⎤‪⎡1 q − 3‬‬
‫⎢ = ‪ ، Ψ ( x ) = τ‬ﻭﺒﺤﺴﺏ ﺍﻟﺒﻨـﺎﺀ ﻟــﹻ‪ Ψ‬ﺍﻟﻤﻌـﺭ‪‬ﻑ‬
‫ﺍﻵﻥ‪ ،‬ﻤﻥ ﺍﻟﻌﻼﻗﺔ ⎥‬
‫⎦‪⎣1 q − 2‬‬
‫ﻥ ﺃﻋﻅﻡ ﺯﻤﺭﺓ‬
‫ﺒﺎﻟﻌﻼﻗﺔ )‪ (٥‬ﻭﻤﻥ ﻜﻭﻥ ) ‪ ، A = Ψ −1 (I‬ﺤﻴﺙ ‪ A‬ﻤﻌﺭ‪‬ﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪ (٢‬ﻨﺠﺩ ﺃ ‪‬‬
‫ﺠﺯﺌﻴﺔ ﻓﻲ ‪ G‬ﺘﻜﻭﻥ ‪ H‬ﻨﻅﺎﻤﻴﺔ ﻓﻴﻬﺎ ﻫﻲ ﻤﻥ ﺍﻟﺸﻜل‪:‬‬
‫‪١١١‬‬
‫ﻴﻌﻘﻭﺏ ﻭ ﻋﻠﻲ ـ ﺩﺭﺍﺴﺔ ﻤﻨﻅﻡ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻭﺫﺍﺕ ﺍﻟﺘﺭﺒﺔ ﺍﻟﻔﺭﺩﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ …‬
‫; ‪M = x, F = x F‬‬
‫‪ G∆F‬ﻤﻌﺭ‪‬ﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪ (١‬ﻭ ‪ x‬ﻤﻌﺭ‪‬ﻑ ﺒﺎﻟﻌﻼﻗﺔ )‪.(١٤‬‬
‫ﻥ‪:‬‬
‫ﺇﺫﺍ ﻤﻥ ﺘﻌﺭﻴﻑ ﺍﻟﻤﻨﻅﻡ ﻭﻤﻥ ﻜﻭﻥ ) ‪ G ⊇ N (H‬ﻓﺈ ‪‬‬
‫‪N (H ) = M = x F‬‬
‫ﻭﻤﻨﻪ‪:‬‬
‫‪= 3 ( p − 1).‬‬
‫‪x .F‬‬
‫‪x ∩F‬‬
‫= ) ‪N (H‬‬
‫ﻨﻅﺭﻴﺔ‪:‬‬
‫ﻟﺘﻜﻥ ﺍﻟﺯﻤﺭﺓ ‪) G‬ﺍﻟﻤﻌﺭ‪‬ﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪ ((٢‬ﻤﻥ ﺍﻟﻤﺭﺘﺒﺔ ‪ n‬ﻭﻟﻴﻜﻥ ‪ r‬ﻗﺎﺴﻤﹰﺎ ﻟــ‪ n‬ﻋﻨﺩﺌﺫ ﻤـﺎ‬
‫ﻴﻠﻲ ﺼﺤﻴﺢ‪:‬‬
‫‪ (١‬ﺘﻭﺠﺩ ﻓﻲ ‪ G‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻭﺍﺤﺩﺓ ﻋﻠﻰ ﺍﻷﻗل ﻤﻥ ﺍﻟﺭﺘﺒﺔ ‪.r‬‬
‫‪ (٢‬ﺠﻤﻴﻊ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻓﻲ ‪ G‬ﻤﻥ ﺍﻟﺭﺘﺒﺔ ‪ r‬ﻤﺘﺭﺍﻓﻘﺔ ﻓﻲ ‪.G‬‬
‫ﺍﻟﺒﺭﻫﺎﻥ‪:‬‬
‫ﻨﺠﺭﻱ ﺍﻻﺴﺘﻘﺭﺍﺀ ﺒﺎﺴﺘﺨﺩﺍﻡ ﻁﺭﻴﻘﺔ ﺍﻻﺴﺘﻘﺭﺍﺀ ﺍﻟﺭﻴﺎﻀﻲ ‪ .n‬ﺇﺫﺍ ﻜﺎﻨﺕ ‪ ،n=١‬ﻋﻨﺩﺌﺫ ﻓـﻲ‬
‫ﻫﺫﻩ ﺍﻟﺤﺎﻟﺔ ﺒﻤﺎ ﺃﻥ ‪ G‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻭﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ GL(q,p‬ﻓﺈﻨﻪ ﻴﻤﻜﻥ ﺩﻭﻤﹰﺎ ﺇﻴﺠﺎﺩ‬
‫ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻭﻗﺎﺒﻠﺔ ﻟﻠﺤل‪ ،‬ﻭﻟﻬﺎ ﺍﻟﺭﺘﺒﺔ ‪ ،n=١‬ﻭﻫﻲ ﻤﻥ ﺍﻟﺸﻜل ‪ ، G G‬ﻭﻤﻥ ﺍﻟﻭﺍﻀﺢ‬
‫ﺃﻨﻬﺎ ﺘﺤﻘﻕ ﺸﺭﻭﻁ ﺍﻟﻨﻅﺭﻴﺔ ﻋﻠﻤﹰﺎ ﺃﻥ ﻋﻨﺼﺭ ﺍﻟﻭﺤﺩﺓ ﻓﻲ ﻫﺫﻩ ﺍﻟﺯﻤﺭﺓ ﻫﻭ ﻋﻨﺼﺭ ﻋﺎﺯل ﻟﻜﻭﻥ‬
‫‪ G‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻭﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ .GL(q,p‬ﻨﻔﺭﺽ ﺃﻥ ﺍﻟﻨﻅﺭﻴﺔ ﺼﺤﻴﺤﺔ ﻤﻥ ﺃﺠـل‬
‫ﻜل ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ‪ ،‬ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل‪ ،‬ﻭﺍﻟﺘﻲ ﺭﺘﺒﺘﻬﺎ ﺃﺼﻐﺭ ﻤﻥ ‪ ،n‬ﻨﺭﻤﺯ ﻷﺼـﻐﺭ‬
‫‪-q‬‬
‫ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻨﻅﺎﻤﻴﺔ ﻓﻲ ‪ G‬ﺒﺎﻟﺭﻤﺯ ‪ ،Q‬ﻓﺘﻜﻭﻥ ‪ Q‬ﺒﺤﺴـﺏ ﺍﻟﺒﻨـﺎﺀ ﻟﻠﺯﻤـﺭﺓ ‪) G‬ﺍﻟﻤﻌﺭﻓـﺔ‬
‫ﺒﺎﻟﻌﻼﻗﺔ )‪ ((٢‬ﻟﻬﺎ ﺍﻟﺸﻜل ﺍﻟﺘﺎﻟﻲ‪:‬‬
‫‪Q = ω I q ; ω q = 1,‬‬
‫) ‪1 ≠ ω ∈ GF ( P‬‬
‫ﺤﻴﺙ ‪ ω‬ﻭ ‪ F‬ﻤﻌﺭ‪‬ﻓﺎﻥ ﺒﺎﻟﻌﻼﻗﺘﻴﻥ )‪ (١‬ﻭ )‪.(٢‬‬
‫‪١١٢‬‬
‫ﻤﺠﻠﺔ ﺠﺎﻤﻌﺔ ﺩﻤﺸﻕ ﻟﻠﻌﻠﻭﻡ ﺍﻷﺴﺎﺴﻴﺔ ـ ﺍﻟﻤﺠﻠﺩ )‪ (١٦‬ـ ﺍﻟﻌﺩﺩ ﺍﻷﻭل ـ ‪٢٠٠٠‬‬
‫‪(A‬‬
‫ﺇﺫﺍ ﻜﺎﻥ ‪ q | r‬ﻓﺒﻤﻭﺠﺏ ﺍﻟﻔﺭﺽ ﺒﺎﻻﺴﺘﻘﺭﺍﺀ ﺘﻭﺠﺩ ﻓﻲ ﺍﻟﺯﻤﺭﺓ‬
‫‪r‬‬
‫ﺠﺯﺌﻴﺔ ‪ B‬ﻋﺎﺯﻟﺔ ﻤﻥ ﺍﻟﺭﺘﺒﺔ‬
‫‪Q‬‬
‫‪q‬‬
‫‪Q‬‬
‫‪ G‬ﺯﻤـﺭﺓ‬
‫‪ ،‬ﺤﻴﺙ ‪ r‬ﻗﺎﺴﻡ ﻤﺎ ﻟــﹻ ‪ G = n‬ﻭﺒﺎﻟﺘﺎﻟﻲ ﺭﺘﺒـﺔ ‪B‬‬
‫ﻫﻲ ‪ r‬ﺃﻱ ‪ ، B = r‬ﻭﻜﺫﻟﻙ ﺍﻟﺯﻤﺭﺓ ‪ B‬ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻭﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓـﻲ ﺍﻟﺯﻤـﺭﺓ ‪G‬‬
‫ﺒﺤﺴﺏ ﺨﻭﺍﺹ ﺍﻟﺯﻤﺭ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﺯﻤﺭ ﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل‪.‬‬
‫ﻥ ﺃﻱ ﺯﻤـﺭﺘﻴﻥ ﺠـﺯﺌﻴﺘﻴﻥ ‪ B1‬ﻭ‬
‫ﻤﻥ ﺍﻟﺴﻬل ﻤﻼﺤﻅﺔ ﺃﻨﹼﻪ ﻀﻤﻥ ﺍﻟﺸﺭﻭﻁ ‪ q | r‬ﺃ ‪‬‬
‫‪ B2‬ﻋﺎﺯﻟﺘﺎﻥ ﻓﻲ ‪ ،G‬ﺤﻴﺙ ‪ B1 = B2 = r‬ﺘﺤﻭﻴﺎﻥ ﺍﻟﺯﻤﺭﺓ ‪ Q‬ﻭﺒﺤﺴﺏ ﺨﻭﺍﺹ ﺍﻟﺯﻤـﺭﺓ‬
‫ﻥ‬
‫ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﺯﻤﺭ ﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻓﺈ ‪‬‬
‫ﻓﻲ‬
‫‪Q‬‬
‫‪Q‬‬
‫‪ B1‬ﻭ‬
‫‪Q‬‬
‫ﻥ‬
‫‪ G‬ﻭﺒﻤﻭﺠﺏ ﺍﻟﻔﺭﺽ ﺒﺎﻻﺴﺘﻘﺭﺍﺀ ﻓﺈ ‪‬‬
‫‪ B2‬ﺯﻤﺭﺘﺎﻥ ﺠﺯﺌﻴﺘﺎﻥ ﻋﺎﺯﻟﺘﺎﻥ ﻭﻗﺎﺒﻠﺘﺎﻥ ﻟﻠﺤل‬
‫‪Q‬‬
‫‪ B1‬ﻭ‬
‫‪Q‬‬
‫‪ B2‬ﻤﺘﺭﺍﻓﻘﺘﺎﻥ ﻓﻲ‬
‫‪Q‬‬
‫‪ G‬ﺒﺤﺴﺏ‬
‫)‪ ،(٢‬ﻭﺒﺎﻟﺘﺎﻟﻲ ‪ B1‬ﻭ ‪ B2‬ﻤﺘﺭﺍﻓﻘﺘﺎﻥ ﻓﻲ ‪.G‬‬
‫‪ (II‬ﺃﻤ‪‬ﺎ ﺇﺫﺍ ﻜﺎﻥ ‪ ، q /| r‬ﻓﻲ ﻫﺫﻩ ﺍﻟﺤﺎﻟﺔ‪ ،‬ﻓﺈﻥ ﺍﻟﺯﻤﺭﺓ ‪ G‬ﺘﺤﻭﻱ ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻤـﻥ‬
‫ﺍﻟﺭﺘﺒﺔ ‪ ،r‬ﺤﻴﺙ ‪ ،r<n‬ﻭﻫﺫﻩ ﺍﻟﺯﻤﺭﺓ ﻤﻭﺠﻭﺩﺓ ﺩﻭﻤﹰﺎ ﺒﺤﺴﺏ ﺍﻟﺒﻨﺎﺀ ﻟﻠﺯﻤﺭﺓ ‪ ،G‬ﻭﻤـﻥ ﺃﺸـﻜﺎﻟﻬﺎ‬
‫ﺇﺤﺩﻯ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻤﻌﺭ‪‬ﻓﺔ ﺒﺎﻟﻌﻼﻗﺔ )‪ ،(١٤‬ﻭﺒﻬﺫﺍ ﻴﻜﻭﻥ ﻗﺩ ﺘ ‪‬ﻡ ﺒﺭﻫﺎﻥ )‪.(١‬‬
‫ﻭﻤﻥ ﺃﺠل ﺒﺭﻫﺎﻥ )‪ (٢‬ﻓﻲ ﺍﻟﺤﺎﻟﺔ ﺍﻟﺘﻲ ﻴﻜﻭﻥ ﻓﻴﻬﺎ ‪ ، q /| r‬ﻟـﺩﻴﻨﺎ ﻤـﻥ ﺃﺠـل ﺃﻱ‬
‫ﻥ ‪ B1‬ﻭ ‪ B2‬ﺴﺘﻜﻭﻨﺎﻥ ﻗﺎﺒﻠﺘﻴﻥ ﻟﻠﺤل ﻓـﻲ‬
‫ﺯﻤﺭﺘﻴﻥ ﺠﺯﺌﻴﺘﻴﻥ ﻋﺎﺯﻟﺘﻴﻥ ‪ B1‬ﻭ ‪ B2‬ﻓﻲ ‪ G‬ﻓﺈ ‪‬‬
‫ﻥ ‪ B1‬ﻭ ‪B2‬‬
‫‪ G‬ﺃﻋﻅﻡ ﺯﻤﺭﺓ ﺠﺯﺌﻴﺔ ﻋﺎﺯﻟﺔ ﻭﻗﺎﺒﻠﺔ ﻟﻠﺤل ﻓﻲ )‪ ،GL(q,p‬ﻭﺒﺤﺴﺏ ]‪ [٣‬ﻓـﺈ ‪‬‬
‫ﺴﺘﻜﻭﻨﺎﻥ ﺯﻤﺭﺘﻴﻥ ﺩﻭﺭﻴﺘﻴﻥ ﻭﺒﺎﻟﺘﺎﻟﻲ ﻓﻬﻤﺎ ﻤﺘﺭﺍﻓﻘﺘﺎﻥ ﻓﻲ ‪ G‬ﻭﻫﻭ ﺍﻟﻤﻁﻠﻭﺏ‪.‬‬
‫‪١١٣‬‬
… ‫ﻴﻌﻘﻭﺏ ﻭ ﻋﻠﻲ ـ ﺩﺭﺍﺴﺔ ﻤﻨﻅﻡ ﺍﻟﺯﻤﺭ ﺍﻟﺠﺯﺌﻴﺔ ﺍﻟﻌﺎﺯﻟﺔ ﻭﺍﻟﻘﺎﺒﻠﺔ ﻟﻠﺤل ﻭﺫﺍﺕ ﺍﻟﺘﺭﺒﺔ ﺍﻟﻔﺭﺩﻴﺔ ﻓﻲ ﺍﻟﺯﻤﺭﺓ‬
REFERENCES
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Invariant Sub Groups in a Group GL(٥,p), which p is a prime number,
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١٩٦٤.
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١٩٦٤.
٥- Johnf.Humphreys, A Course in Group Theory, Oxford, New York, Tokyo,
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٦- Michale Aschbacher, Finite Group Theory, Cambridge, New York, Sydney,
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