Order Of an Element

Duration: 6 min

This video lesson is available to enrolled students.

Return to /learn/ISRO/discrete-mathematics/group-theory/determining-the-order-of-elements-within-groups/asset-order-of-an-element after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. General Aptitude & Reasoning: Number System, Percentage, Ratio and Proportion (Ratios), Average, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Speed Time and Distance, Mixture and Alligation, Data Interpretation, Algebra, Probability, Permutation and Combination, Venn Diagram, HCF LCM, Divisibility Rules, Age Problems, Pipe and Cistern, Partnership, Linear Algebra, Sequence and Series (Series), Statistics, Powers and Exponents (Surds and Indices), Arithmetic, Mensuration and Geometry, Calculus, Trigonometry, Vectors, Speed Mathematics and Basic Mathematics, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Analogy, Series (Number and Letter Series) (Numerical Relations and Reasoning), Puzzle Solving (Puzzles), Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Calendar, Seating Arrangements, Coding Decoding, Blood Relations, Directions (Direction Test), Clock, Analytical Reasoning (Counting Figures Reasoning), Ranking, Order and Sequence, Logical Sequence Of Words, Inequality, Data Sufficiency, Cubes & Dices
  3. Computer Networks & Cybersecurity: 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 Security
  4. Programming, Data Structures & Algorithms: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  5. AI, Machine Learning, Big Data & Cloud: Emerging Tech Trends, Artificial Intelligence, Big Data Systems, Approaches to AI, Search Algorithms, Knowledge Representation, Cloud Computing
  6. Digital Logic & Computer Organization: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer
  7. TOC & Compiler: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory, 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
  8. Discrete Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic
  9. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Distributed Systems
  10. Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, Object Oriented Technology, HTML Basics, JavaScript-Basics, Java, XML
  11. Database Management System: 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, Database Recovery, Data Warehousing & Mining, Enhanced Data Models, NoSQL
  12. Other Recurring CS Topics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, Multimedia
  13. Live Classes: ISRO Live classes
  14. Full Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video lecture provides a comprehensive introduction to the concept of the "Order of an element" within the context of group theory. It begins by defining the order of an element 'a' in a group (A, *) as the smallest positive integer 'n' such that a^n equals the identity element 'e', denoted as O(a). The instructor outlines four key properties: the identity element always has an order of one; an element and its inverse share the same order; and in infinite groups, the order is infinite, except for the identity. The lecture then transitions to a practical application, posing a question to find the order of each element in the group {0, 1, 2, 3} under addition modulo 4 (+4). The instructor systematically constructs the Cayley table for this group and calculates the order for every element, demonstrating the theoretical concepts with concrete arithmetic.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The session opens with a slide titled "Order of an element," defining the notation O(a) for an element 'a' in a group (A, *). The instructor explains the first property: O(a) = n, where 'n' is the smallest positive integer such that a^n = e. He lists three additional properties: the order of the identity element is always one; the order of an element and its inverse is always the same; and in an infinite group, the order does not exist or is infinite, except for the identity. Around the 1:50 mark, the slide changes to a specific problem: "Q consider a group {0,1,2,3}, +4 and find the order of each element?" A blank Cayley table for the operation +4 is displayed, setting the stage for the practical demonstration.

  2. 2:00 – 5:00 02:00-05:00

    The instructor begins filling out the Cayley table for the group {0, 1, 2, 3} under addition modulo 4. He fills the first row and column with 0, 1, 2, 3 respectively. He then calculates the remaining entries: 1+1=2, 1+2=3, 1+3=0; 2+1=3, 2+2=0, 2+3=1; and 3+1=0, 3+2=1, 3+3=2. With the table complete, he starts calculating the order of element 0. He writes 0^1 = 0, noting that since 0 is the identity, its order is 1. Next, he calculates the order of element 1. He writes 1^1 = 1, 1^2 = 1+1=2, 1^3 = 1+2=3, and 1^4 = 1+3=0. Since 1^4 equals the identity 0, he concludes that the order of 1 is 4.

  3. 5:00 – 6:24 05:00-06:24

    Continuing the calculation, the instructor determines the order of element 2. He writes 2^1 = 2 and 2^2 = 2+2=0. Since 2^2 equals the identity 0, he concludes O(2) = 2. He then moves to element 3, writing 3^1 = 3, 3^2 = 3+3=2, 3^3 = 3+2=1, and 3^4 = 3+1=0. Because 3^4 equals the identity 0, he determines O(3) = 4. The instructor summarizes the findings on the board: O(0)=1, O(1)=4, O(2)=2, and O(3)=4. He points out that elements 1 and 3 are generators of the group because their order equals the total number of elements in the group. The video concludes with the completed table and all calculated orders visible.

The lecture effectively bridges theoretical definitions with practical computation. By defining the order of an element as the smallest power yielding the identity, the instructor establishes a clear criterion for calculation. The example of the group Z4 under addition modulo 4 serves as a perfect illustrative case. The step-by-step construction of the Cayley table ensures that the arithmetic operations are transparent. The calculation of orders for 0, 1, 2, and 3 reinforces the properties discussed earlier, specifically showing that the identity has order 1, elements can have orders less than the group size (like 2), and elements can have orders equal to the group size (like 1 and 3). This progression from definition to property to application provides a complete understanding of how to find the order of elements in a finite group.

Loading lesson…