Cyclic group- group theory
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This document defines cyclic groups and discusses their order theorem. It explains that a cyclic group G is generated by a single element a, such that every element of G can be expressed as an power of a. It then proves that the order of a cyclic group equals the order of its generator using the division algorithm, showing that for any integer m, m can be expressed as nq + r, where n is the order of the generator and 0 <= r < n. It provides examples of applications of group theory in fields like physics, biology, crystal structure analysis, and coding theory.
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This document discusses group theory and symmetry elements as they relate to several different molecules. It provides examples of identifying the point groups, symmetry elements like rotation axes and planes of inversion, and determining the representations of atomic and molecular orbitals for molecules like ammonia, acetone, ethanediol, propadiene, water, BH3, cyclopropenyl cation, butadiene, and trichlorborane. Worked examples are provided to demonstrate how to analyze symmetry properties and construct molecular orbital diagrams for various systems.
Vedic maths .PPT
Vedic mathematics is a system of mathematics consisting of 16 sutras or aphorisms obtained from ancient Hindu scriptures called the Vedas. It was presented in the early 20th century by Bharati Krishna Tirthaji Maharaja, an Indian scholar. The sutras provide concise formulae for solving problems through unique techniques like vertically-and-crosswise calculations without needing multiplication tables beyond 5x5. Some examples include techniques for squaring numbers and multiplying multi-digit numbers mentally through a carry-over method. Vedic mathematics was applied in areas like astronomy, astrology and constructing calendars.
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Cyclic group- group theory
- 1. CYCLIC GROUP and ORDER THEOREM - By Ayush Agrawal
- 2. APPLICATIONS OF GROUP THEORY
- 3. APPLICATIONS OF GROUP THEORY • Physical sciences • Biological Sciences • Study of Crystal Structure • Configuration of molecules • Structure of human Genes • And in Coding theory
- 4. GROUP A non-empty set G is called a group if it is defined by binary operation * It must also follow these conditions: 1. Closure property 2. Associative law 3. Existence of identity 4. Existence of inverse
- 5. Now let us come to the point CYCLIC GROUP
- 6. CYCLIC GROUP Definition: A group G is said to be cyclic if for some a in G, every element x in G can be expressed as a^n, for some integer n. Thus G is Generated by a i.e. G=(a) Now let us study why order of cyclic group equals order of its generator.
- 7. Note: • For some time we must consider that the group we are taking has MULTIPLICATION as its binary operation. • Though any other operation is equally suitable.
- 9. Division algorithm • 9= 4x2+1 • 12= 7x1+5 • 15= 3x5+0 Similarly, m= nq+r, where 0<=r<n
- 11. Hence Prooved
- 12. Now lets solve a problem on this
- 13. I= -I Set of integers is equal to itself even when multiplied by -1
- 15. Any
- 16. THE END
Editor's Notes
- , and n=o(a)
- Now lets solve a problem on this
- Set of integers is equal to itself even when multiplied by -1
- Here’s the example
- Any
- THE END