Generating Element and Cyclic Group

Duration: 4 min

This video lesson is available to enrolled students.

Return to /learn/COAL-INDIA-MANAGEMENT-TRAINEE/engineering-maths/group-theory/determining-the-order-of-elements-within-groups/asset-generating-element-and-cyclic-group after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. Paper-2 | Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  3. Paper-2 | Databases / DBMS & SQL: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control
  4. Paper-2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  5. Paper-2 | Computer Networks & Security: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Network Access Technologies, Network Security
  6. Paper-2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  7. Paper-2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  8. Paper-2 | Theory of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines
  9. Paper-2 | Computer Organization & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer
  10. Paper-2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  11. Paper-1 | General Awareness: Polity, Physics, Biology, Chemistry, Static General Knowledge, GK- Current Affairs, History, Geography, Indian Economy, Environment
  12. Paper-1 | Logical Reasoning: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Data Interpretation, Blood Relations, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Clock, Calendar, Cubes & Dices, Directions (Direction Test), Analogy, Seating Arrangements, Analytical Reasoning (Counting Figures Reasoning)
  13. Paper-1 | Quantitative Aptitude: Ratio and Proportion (Ratios), Mixture and Alligation, Permutation and Combination, Number System, HCF LCM, Speed Time and Distance, Average, Age Problems, Percentage, Profit and Loss, Time and Work, Pipe and Cistern, Simple Interest and Compound Interest, Powers and Exponents (Surds and Indices), Mensuration and Geometry, Probability, Statistics, Arithmetic, Algebra
  14. Paper-1 | English: Vocabulary, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Cloze Test (Cloze Passage)
  15. Coal India Previous Year Paper:
  16. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus
  17. Coal India Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video lecture introduces fundamental concepts in abstract algebra, specifically focusing on generating elements and cyclic groups. The instructor begins by defining a generating element as an element whose integral powers can represent every element in a set. The lecture then progresses to defining cyclic groups and establishing their key properties, such as the relationship between a generator and its inverse. To illustrate these abstract concepts, the instructor works through concrete examples using the fourth and third roots of unity, calculating the order of each element using Cayley tables. Finally, the lesson concludes by applying these principles to a multiple-choice question from a GATE-2009 exam involving a composition table.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a definition slide titled 'Generating element or Generator'. The on-screen text states: 'A element 'a' is said to be a generating element, if every element of A is an integral power of a, i.e. every element of A can be represented using power of a.' The instructor underlines key phrases like 'generating element' and 'integral power of a' to emphasize the definition. He displays the set notation A = {a^1, a^2, a^3, a^4, a^5....} to visually represent the powers of the generator 'a'. The instructor explains that if a single element can generate the entire set through its powers, it is a generator.

  2. 2:00 – 4:10 02:00-04:10

    The lecture transitions to practical examples. First, a question asks to consider the set of fourth roots of unity {-1, 1, i, -i} and find the order of each element. A Cayley table is displayed, and the instructor calculates orders: O(1)=1, O(-1)=2, O(i)=4, and O(-i)=4, writing out powers like i^4=1. Next, he analyzes the cube roots of unity {1, w, w^2}, showing a table where O(w)=3 and O(w^2)=3. He then defines a 'Cyclic group' as a group containing at least one generator. Two properties are listed: 1) If an element is a generator, its inverse is also a generator (illustrated with w <-> w^2 and i <-> -i). 2) The order of a cyclic group is always the order of the generating element. The segment ends with a GATE-2009 problem showing a composition table for {a, b, c, d} and asking to identify the generators from multiple-choice options.

The video effectively bridges theoretical definitions with practical application. It starts by establishing the rigorous definition of a generator, requiring every element to be an integral power of a single element. This concept is then operationalized through the analysis of roots of unity, where the instructor demonstrates how to determine element orders using Cayley tables. The distinction between the order of an element and the order of the group is clarified through the property that the group order equals the generator's order. The lesson culminates in a problem-solving scenario, testing the student's ability to identify generators within a given composition table, reinforcing the theoretical properties discussed earlier.

Loading lesson…