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ধরা যাক p একটি মৌলিক সংখ্যা। ধরা যাক G একটি গোষ্ঠী যেমন প্রতিটি g ∈ G-এর জন্য একটি n ∈ ℕ বিদ্যমান থাকে, যেমন gpn = 1। নিম্নলিখিত বিবৃতিগুলির মধ্যে কোনটি মিথ্যা?

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CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. যদি |G| = p6 হয়, তাহলে G-এর p2 সূচক সহ একটি উপগোষ্ঠী রয়েছে।
  2. যদি |G| = p6 হয়, তাহলে G-এর অন্তত পাঁচটি স্বাভাবিক উপগোষ্ঠী রয়েছে।
  3. G-এর কেন্দ্র অসীম হতে পারে।
  4. |G| = p6 সহ একটি G বিদ্যমান থাকে যাতে G-এর ঠিক ছয়টি স্বাভাবিক উপগোষ্ঠী রয়েছে।

Answer (Detailed Solution Below)

Option 4 : |G| = p6 সহ একটি G বিদ্যমান থাকে যাতে G-এর ঠিক ছয়টি স্বাভাবিক উপগোষ্ঠী রয়েছে।
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Detailed Solution

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ধারণা:

(i) সাইলোর প্রথম উপপাদ্য: ধরা যাক G একটি সসীম গোষ্ঠী এবং p একটি মৌলিক সংখ্যা। যদি pk |G| কে ভাগ করে, তাহলে G-এর অন্তত একটি pk ক্রমের উপগোষ্ঠী রয়েছে।

(ii) যদি f o(G) = pn হয় তাহলে অন্তত n - 1 টি স্বাভাবিক উপগোষ্ঠী বিদ্যমান থাকে pr এর জন্য যেখানে 0 ≤ r ≤ n

ব্যাখ্যা:

(1): প্রদত্ত |G| = p6, সুতরাং p4 p6 কে ভাগ করে। তাহলে সাইলোর প্রথম উপপাদ্য অনুসারে, একটি উপগোষ্ঠী H ⊂ G বিদ্যমান থাকে যেমন |H| = p4

সুতরাং H এর সূচক = p6/ p4 = p2
বিকল্প (1) সত্য

(2): |G| = p6 তাহলে ফলাফল (ii) অনুসারে G এর অন্তত পাঁচটি স্বাভাবিক উপগোষ্ঠী রয়েছে

বিকল্প (2) সত্য এবং বিকল্প (4) মিথ্যা

(3): যদি আমরা (P\((\mathbb N), \triangle\)) এবং p = 2 বিবেচনা করি তবে G এর কেন্দ্র অসীম।
বিকল্প (3) সত্য
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