Finite Group Practice Question

Duration: 4 min

This video lesson is available to enrolled students.

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  1. About Course: UPPSC Course Overview
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  3. Paper 1 | General Hindi: हिन्दी की रूप रचना, वर्णमाला, समास, बहु-विषयक सामग्री (विभिन्न अवधारणाओं को जोड़ने वाली सामग्री), हिन्दी की ध्वनि और स्वनिम व्यवस्था, प्राचीन और मध्यकालीन भारतीय आर्यभाषाएँ, काव्य के लक्षण, हेतु और प्रयोजन, हिन्दी शब्द रचना, हिन्दी साहित्येतिहास दर्शन, भक्तिकालीन काव्यधाराएँ, भारतीय काव्यशास्त्र की परंपरा, भारतीय नवजागरण और स्वाधीनता आंदोलन, अन्य गद्य विधाएँ एवं प्रवासी साहित्य, निर्धारित उपन्यासों का तुलनात्मक अध्ययन, गोदान, हिन्दी कहानी का विकास, स्वातंत्र्योत्तर और नयी कहानी, हिन्दी निबंध का विकास, शुक्लोत्तर निबंध, द्विवेदी और शुक्ल युग के निबंध, हिन्दी की संवैधानिक स्थिति, आधुनिक भारतीय आर्यभाषाएँ और हिन्दी का उद्भव, हिन्दी भाषा प्रयोग के विविध रूप, रीतिकाल, जीवनी, द्विवेदी युग और छायावाद, रीति, वक्रोक्ति और औचित्य सिद्धान्त, शब्दशक्ति और ध्वनि सिद्धान्त, रस सिद्धान्त और साधारणीकरण, उत्तर आधुनिकतावाद, लोहिया दर्शन, हिन्दी उपन्यास का विकास, हिन्दी नवजागरण, अंबेडकर दर्शन, आदिकाल, प्रयोगवाद और नयी कविता, छायावादी काव्य: प्रसाद और निराला, प्रगतिवाद से समकालीन कविता, निर्गुण भक्ति काव्य, द्विवेदीयुगीन काव्य, रीतिकालीन काव्य, आरंभिक हिन्दी उपन्यास, स्वातंत्र्योत्तर उपन्यास, आत्मकथा, स्वाधीनता-पूर्व प्रेमचंदोत्तर उपन्यास, निर्धारित कहानियों का तुलनात्मक अध्ययन, साठोत्तरी नाटक, प्रसाद और प्रसादोत्तर युग के नाटक, भारतेन्दु युगीन नाटक, स्वातंत्र्योत्तर नाटक, भारतेन्दु युगीन निबंध, संस्मरण और रेखाचित्र, व्यंग्य, एक साहित्यिक की डायरी, संस्कृति के चार अध्याय, सगुण भक्ति काव्य, हिन्दी नाटक, एकांकी और रंगमंच, आधुनिक गद्य और हिन्दी नवजागरण, अलंकार सिद्धान्त, हिन्दी आलोचना का विकास, भक्ति आंदोलन, कल्पना, बिम्ब, प्रतीक, मिथक और फंतासी, नयी कविता का विस्तार और साठोत्तरी कविता, उत्तर-छायावादी और प्रगतिवादी काव्य, काव्यगुण और काव्यदोष, प्लेटो, अरस्तू और पाश्चात्य परंपरा, मार्क्सवाद, स्वच्छन्दतावादी काव्यशास्त्र, आधुनिक पाश्चात्य आलोचना, भाषा और भाषा परिवार, देवनागरी लिपि
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  12. Paper 1 | Discrete Structures and Theory of Logic: Permutation, Combination and Probability / क्रमचय, संचय एवं प्रायिकता, Set Theory, Relations, Functions, Propositional and Predicate Logic, Graph Theory, Group Theory
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  17. Paper 2 | Image Processing: Digital Image Fundamentals, Image Enhancement, Image Restoration, Color Image Processing, Image Compression
  18. Paper 2 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  19. Paper 2 | Computer Graphics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing
  20. Paper 2 | Web Technologies: Website, Browser & Services, Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, Emerging Technologies, HTML Basics, JavaScript-Basics, Java
  21. Paper 2 | Soft Computing: Fuzzy Sets, Artificial Neural Networks, Genetic Algorithms
  22. Paper 2 | Distributed System: Distributed Systems
  23. UPPSC Live Class Recordings: UPPSC Polytechnic Lecturer 2025 (CS)
  24. Previous Year Papers:
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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video is a lecture on identifying finite groups from a list of algebraic structures. The instructor systematically evaluates 14 options, using Cayley tables to check for group axioms like closure, identity, and inverses. He marks valid groups and crosses out invalid ones, providing reasoning for each decision based on the properties of the set and operation.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor begins by analyzing options 1 through 8, using red ink to mark valid or invalid structures. He starts with option 1, the set {0} under addition. He fills the table to show 0+0=0, confirming it is a group. Next, he analyzes option 2, {0} under multiplication. He crosses it out with a red X, explaining that the identity element for multiplication is 1, which is not present in the set. For option 3, the set {1} under addition, he calculates 1+1=2. Since 2 is not in the set, closure is violated, and he eliminates it. Option 4, {1} under multiplication, is accepted as a group because 1*1=1, satisfying all axioms. He then looks at option 5, {0, 1} under addition. He fills the table and finds 1+1=2, which is outside the set, so he crosses it out. Option 6, {0, 1} under multiplication, is rejected because 0 has no multiplicative inverse. Options 7 and 8, involving the set {-1, 0, 1} with addition and multiplication, are also eliminated. Addition fails closure (1+1=2), and multiplication fails because 0 has no inverse.

  2. 2:00 – 4:29 02:00-04:29

    In the second segment, the instructor moves to options 9 through 14. Option 9, {-1, 1} under addition, is crossed out because -1+1=0, which is not in the set. Option 10, {-1, 1} under multiplication, is validated. The table shows closure, identity (1), and inverses (-1 is its own inverse). Option 11, {-2, -1, 0, 1, 2} under addition, is rejected because 2+1=3 is outside the set. Option 12, the same set under multiplication, is rejected because 0 has no inverse. Finally, he discusses options 13 and 14. Option 13, {1, w, w^2} under multiplication, is identified as the group of cube roots of unity, noting w^3=1. Option 14, {-1, 1, i, -i} under multiplication, is identified as the group of fourth roots of unity. Both are marked as valid finite groups.

The lecture focuses on identifying finite groups by testing algebraic structures against group axioms. The instructor uses Cayley tables to check for closure, identity elements, and inverses. He systematically eliminates options that fail these criteria, such as sets lacking an identity element (like {0} under multiplication) or sets failing closure (like {1} under addition where 1+1=2). The valid examples provided include trivial groups like {0} under addition and {1} under multiplication, as well as more complex structures like the multiplicative group of roots of unity. This process highlights that finite sets of integers under addition rarely form groups due to closure issues, while specific multiplicative sets often do. The key takeaway is that not all sets with operations form groups; specific structural properties must be met. The instructor emphasizes checking the identity element first, as its absence immediately disqualifies a structure. He also points out that 0 is a common point of failure in multiplicative groups because it lacks an inverse.

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