Order Of an Element

Duration: 6 min

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Module outline

  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
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  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. 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, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: 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, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. 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, Complexity Theory
  10. 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
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  13. Full Length Mock Tests:
  14. Sectional Tests:
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.

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