16650

1799 دكتري

دكتري

رياضي محض(نظريه گروه)

اصفهان : دانشگاه صنعتي اصفهان

1400

هفت، 53ص.

بيژن طائري

عليرضا عبدالهي

1400/07/25

كتابنامه

رياض

رياضي

1400/07/26

2738998

ﻓﺮﺽ ﮐﻨﻴﻢ G ﻳﮏ ﮔﺮﻭﻩ ﻣﺘﻨﺎﻫﯽ ﻣﺘﻤﺎﻳﺰ ﺑﺎ ﻫﺮ ﮔﺮﻭﻩ ﺩﻭﺭﯼ ﺍﺯ ﻣﺮﺗﺒﻪﯼ ﺗﻮﺍﻧﯽ ﺍﺯ ﻳﮏ ﻋﺪﺩ ﺍﻭﻝ ﺑﺎﺷﺪ. ﻣﺠﻤﻮﻋﻪﯼ ﺭﺋﻮﺱ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ G))∆ ﺑﺮﺍﺑﺮ ﺑﺎ ﻣﺠﻤﻮﻋﻪﯼ ﺗﻤﺎﻡ ﺯﻳﺮﮔﺮﻭﻩﻫﺎﯼ ﺳﺮﻩﯼ G ﺍﺳﺖ ﮐﻪ ﺩﺭ ﺯﻳﺮﮔﺮﻭﻩ ﻓﺮﺍﺗﻴﻨﯽ G ﻗﺮﺍﺭ ﻧﺪﺍﺭﻧﺪ ﻭ ﺩﻭ ﺭﺃﺱ ﻣﺘﻤﺎﻳﺰ H ﻭ K ﻣﺠﺎﻭﺭﻧﺪ ﻫﺮﮔﺎﻩ .G= ﺩﺭ ﺍﻳﻦ ﺭﺳﺎﻟﻪ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﺭﺍ ﻣﻄﺎﻟﻌﻪ ﻣﯽﮐﻨﻴﻢ ﻭ ﻧﺘﺎﻳﺠﯽ ﺩﺭ ﺍﺭﺗﺒﺎﻁ ﺑﺎ ﺗﺎﺛﻴﺮ ﺁﻥ ﺩﺭ ﺳﺎﺧﺘﺎﺭ ﮔﺮﻭﻩ ﺭﺍ ﺛﺎﺑﺖ ﻣﯽﮐﻨﻴﻢ. ﮔﺮﻭﻩﻫﺎﻳﯽ ﮐﻪ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﺁﻥﻫﺎ ﺩﺍﺭﺍﯼ ﻋﺪﺩ ﻏﻠﺒﻪ 1 ﻭ 2 ﺍﺳﺖ ﺭﺍ ﺭﺩﻩﺑﻨﺪﯼ ﻣﯽﮐﻨﻴﻢ. ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺭﺩﻩﺑﻨﺪﯼ ﮔﺮﻭﻩﻫﺎﯼ ﺑﺎ ﻋﺪﺩ ﻏﻠﺒﻪ 1 ﻧﺸﺎﻥ ﻣﯽﺩﻫﻴﻢ اگر گراف الحاق يك گروه با گراف الحاق گروه متناوب A_4 يكريخت باشد آنگاه دو گروه يكريخت خواهند شد. ﺛﺎﺑﺖ ﻣﯽﮐﻨﻴﻢ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ ﺑﺮﺍﯼ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﻣﺘﻨﺎﻇﺮ ﺑﺎ ﮔﺮﻭﻩ ﻣﺘﻨﺎوب A_5, ﺑﺮﺍﺑﺮ ﺑﺎ 15 ﺍﺳﺖ. ﺩﺭ ﻭﺍﻗﻊ ﻧﺸﺎﻥ ﻣﯽﺩﻫﻴﻢ ﮐﻪ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ ﺑﺮﺍﯼ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﻣﺘﻨﺎﻇﺮ ﺑﺎ ﮔﺮﻭﻩﻫﺎﯼ ﺳﺎﺩﻩﯼ ﻧﺎﺁﺑﻠﯽ ﻣﺘﻨﺎﻫﯽ ﺑﺰﺭﮔﺘﺮ ﺍﺯ 15 ﺍﺳﺖ. ﻫﻤﭽﻨﻴﻦ ﺍﮔﺮ G ﮔﺮﻭﻩ ﺳﺎﺩﻩﯼ ﻧﺎﺁﺑﻠﯽ ﻣﺘﻨﺎﻫﯽ ﺑﺎﺷﺪ ﺑﻪ ﻃﻮﺭﯼ ﮐﻪ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﻣﺘﻨﺎﻇﺮ ﺁﻥ ﺑﺮﺍﺑﺮ ﺑﺎ 15 ﺑﺎﺷﺪ، ﺁﻥﮔﺎﻩ ﺛﺎﺑﺖ ﻣﯽﮐﻨﻴﻢ G ﺑﺎ A_5 ﻳﮑﺮﻳﺨﺖ ﺍﺳﺖ. ﻗﺒﻼً ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﺍﮔﺮ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﻣﺘﻨﺎﻇﺮ ﺑﺎ ﻳﮏ ﮔﺮﻭﻩ ﮐﻮﭼﮑﺘﺮ ﻳﺎ ﻣﺴﺎﻭﯼ 7 ﺑﺎﺷﺪ، ﺁﻥﮔﺎﻩ ﮔﺮﻭﻩ ﺣﻞﭘﺬﻳﺮ ﺍﺳﺖ. ﺍﻳﻦ ﮐﺮﺍﻥ ﺭﺍ ﺑﻬﺘﺮ ﮐﺮﺩﻩ ﻭ ﺛﺎﺑﺖ ﻣﯽﮐﻨﻴﻢ ﮐﻪ ﺍﮔﺮ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﮐﻮﭼﮑﺘﺮ ﻳﺎ ﻣﺴﺎﻭﯼ 14 ﺑﺎﺷﺪ، ﺁﻥﮔﺎﻩ ﮔﺮﻭﻩ ﺣﻞﭘﺬﻳﺮ ﺍﺳﺖ ﻭ ﺍﻳﻦ ﺑﻬﺘﺮﻳﻦ ﮐﺮﺍﻥ ﺍﺳﺖ. ﺩﺭ ﻭﺍﻗﻊ A_5 ﻣﺜﺎﻟﯽ ﺍﺯ ﻳﮏ ﮔﺮﻭﻩ ﺣﻞﻧﺎﭘﺬﻳﺮ ﺍﺳﺖ ﮐﻪ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﺁﻥ ﺩﺍﺭﺍﯼ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ 15 ﺍﺳﺖ. ﺍﮔﺮ G ﮔﺮﻭﻩ ﺣﻞﻧﺎﭘﺬﻳﺮ ﻣﺘﻨﺎﻫﯽ ﺑﺎﺷﺪ ﺑﻪ ﻃﻮﺭﯼ ﮐﻪ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﺁﻥ 15 ﺑﺎﺷﺪ، ﺁﻥﮔﺎﻩ ﻧﺸﺎﻥ ﻣﯽﺩﻫﻴﻢ G)) G/Φ با A_5 يكريخت است. ﺩﺭ ﺭﺍﺑﻄﻪ ﺑﺎ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ ﺑﺮﺍﯼ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ، ﮔﺮﻭﻩﻫﺎﻳﯽ ﮐﻪ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﺁﻥﻫﺎ ﺩﺍﺭﺍﯼ ﻋﺪﺩ ﺍﺳﺘﻘﻼﻝ ﮐﻮﭼﮑﺘﺮ ﻳﺎ ﻣﺴﺎﻭﯼ 3 ﺍﺳﺖ ﺭﺍ ﺭﺩﻩﺑﻨﺪﯼ ﻣﯽﮐﻨﻴﻢ. ﺩﺭ ﺍﺩﺍﻣﻪ ﺑﺮﺭﺳﯽ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﻧﺘﺎﻳﺞ ﻣﻬﻤﯽ ﺭﺍ ﺩﺭ ﺍﺭﺗﺒﺎﻁ ﺑﺎ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﮔﺮﻭﻩﻫﺎﯼ ﺩﻭﺭﯼ ﻭ ﮔﺮﻭﻩ A_5 ﺑﺪﺳﺖ ﻣﯽﺁﻭﺭﻳﻢ. ﺛﺎﺑﺖ ﻣﯽﮐﻨﻴﻢ ﺍﮔﺮ G ﻳﮏ ﮔﺮﻭﻩ ﻣﺘﻨﺎﻫﯽ ﺩﻟﺨﻮﺍﻩ ﺑﺎﺷﺪ ﺑﻪ ﻃﻮﺭﯼ ﮐﻪ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﺁﻥ ﺑﺎ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﮔﺮﻭﻩ ﺩﻭﺭﯼ H ﻳﮑﺮﻳﺨﺖ ﺑﺎﺷﺪ، ﺁﻥﮔﺎﻩ G ﺩﻭﺭﯼ ﺍﺳﺖ. ﺗﻮﺟﻪ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻴﻢ G ﻭ H ﻟﺰﻭﻣﺎ ﻳﮑﺮﻳﺨﺖ ﻧﻴﺴﺘﻨﺪ. ﻋﻼﻭﻩ ﺑﺮ ﺁﻥ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺷﮑﻞ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﻳﮏ ﮔﺮﻭﻩ ﻣﯽﺗﻮﺍﻥ ﺩﻭﺭﯼ ﺑﻮﺩﻥ ﻳﺎ ﻧﺒﻮﺩﻥ ﮔﺮﻭﻩ ﺭﺍ ﻣﺸﺨﺺ ﮐﺮﺩ. ﻫﻤﭽﻨﻴﻦ ﻧﺸﺎﻥ ﻣﯽﺩﻫﻴﻢ ﮐﻪ ﺍﮔﺮ ﺩﻭ ﮔﺮﻭﻩ ﺩﺍﺭﺍﯼ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﻳﮑﺴﺎﻥ ﺑﺎﺷﻨﺪ ﻭ ﻳﮑﯽ ﺍﺯ ﮔﺮﻭﻩﻫﺎ ﺁﺑﻠﯽ يا ﭘﻮﭺﺗﻮﺍﻥ ﺑﺎﺷﺪ، ﺁﻥﮔﺎﻩ ﻟﺰﻭﻣﺎ به ترتيب ﮔﺮﻭﻩ ﺩﻳﮕﺮ ﺁﺑﻠﯽ يا ﭘﻮﭺﺗﻮﺍﻥ ﻧﻴﺴﺖ. ﺑﻪ ﻋﻨﻮﺍﻥ ﻳﮏ ﻧﺘﻴﺠﻪ ﻣﻬﻢ ﺩﺭ ﺑﺮﺭﺳﯽ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ، ﺛﺎﺑﺖ ﻣﯽﮐﻨﻴﻢ ﺍﮔﺮ G ﻳﮏ ﮔﺮﻭﻩ ﻣﺘﻨﺎﻫﯽ ﺑﺎﺷﺪ ﺑﻪ ﻃﻮﺭﯼ ﮐﻪ گراف الحاق آ با گراف الحاق A_5 يكريخت باشد، آنگاه G با A_5 خواهد بود. و در پايان ﮔﺮﻭﻩﻫﺎﻳﯽ ﮐﻪ ﮔﺮﺍﻑ ﺍﻟﺤﺎﻕ ﺁﻥﻫﺎ ﻓﺎﻗﺪ ﺯﻳﺮﮔﺮﺍﻑ 3,3K، ﮔﺮﺍﻑ ﺩﻭﺑﺨﺸﯽﮐﺎﻣﻞ ﺑﺎ ﺑﺨﺶﻫﺎﯼ ﺑﺎ ﺍﻧﺪﺍﺯﻩ 3، ﺭﺍ ﺭﺩﻩﺑﻨﺪﯼ ﻣﯽﮐﻨﻴﻢ.

Let G be a finite group which is not isomorphic with any cyclic p-group (p is a prime). The join graph ∆(G) is an undirected simple whose vertices are the proper subgroups of G, which are not contained in the Frattini subgroup Φ(G) of G and two vertices H and K are joined by an edge if and only if G =. We classify finite groups whose join graphs have domination number ≤ 2 and independence number ≤ 3. We show that ∆(G) and ∆(A4) are isomorphic if and only if G and A4 are isomorphic. Also we show that if the independence number of ∆(G) is less than 15, then G is solvable; moreover if the equality holds and G is non-solvable, then G/Φ(G) and A5 are isomorphic. Among other results, we show that if G is a finite cyclic group and H is a finite group such that ∆(G) and ∆(H) are isomorphic, then H is cyclic. Also we prove that ∆(G) are isomorphic ∆(A5) if and only if G and A5 are isomorphic . Finally we classify all finite groups with K3,3-free graph

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عليرضا عبدالهي