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Concept of Permutation Group Video Lecture | Mathematics for Competitive Exams
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FAQs on Concept of Permutation Group Video Lecture - Mathematics for Competitive Exams
| 1. What is a permutation group? |  |
Ans. A permutation group is a mathematical concept that consists of a set of permutations together with a binary operation called composition. It forms a group structure where permutations can be combined or composed to create new permutations.
| 2. How are permutation groups useful in mathematics? | |
Ans. Permutation groups have various applications in mathematics, particularly in the field of group theory. They are used to study the symmetry of objects, such as geometric figures, molecular structures, and mathematical equations. Permutation groups also provide a framework for analyzing permutations and their properties, which has implications in cryptography, combinatorics, and algebraic structures.
| 3. Can you give an example of a permutation group? | |
Ans. Yes, a common example of a permutation group is the symmetric group, denoted as S_n, which consists of all possible permutations of n elements. For instance, if we have three elements {1, 2, 3}, the symmetric group S_3 would include permutations like (1, 2, 3), (3, 2, 1), (1, 3, 2), etc. These permutations can be combined using composition to form new permutations within the group.
| 4. What are the properties of a permutation group? | |
Ans. A permutation group has several important properties. Firstly, it always contains the identity permutation, which leaves all elements unchanged. Secondly, for every permutation in the group, its inverse permutation is also present. Additionally, the composition of two permutations in the group results in another permutation within the same group. Finally, a permutation group is closed under composition, meaning that the composition of any two permutations in the group remains within the group.
| 5. How can permutation groups be represented? | |
Ans. Permutation groups can be represented in different ways, depending on the context. One common representation is through cycle notation, where each permutation is expressed as a product of disjoint cycles. Another representation is through permutation matrices, which use matrices to represent permutations. Additionally, permutation groups can be represented using generators and relations, where a set of generators is defined along with specific relations that govern their composition.
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Dec 23, 2024 Last updated |
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