GRE Exam > GRE Videos > Mathematics for GRE Paper II > Examples of Group & Abelian Group
Examples of Group & Abelian Group Video Lecture | Mathematics for GRE Paper II
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FAQs on Examples of Group & Abelian Group Video Lecture - Mathematics for GRE Paper II
| 1. What is a group in mathematics? |  |
Ans. In mathematics, a group is a set equipped with an operation that combines any two elements of the set to form a third element. This operation must satisfy four properties: closure, associativity, identity, and invertibility. Groups are widely studied in various branches of mathematics and have applications in many areas, including algebra, geometry, and physics.
| 2. What is an Abelian group? | |
Ans. An Abelian group, also known as a commutative group, is a group in which the operation is commutative. This means that for any two elements of the group, their order of combination does not affect the result. In other words, if a and b are elements of an Abelian group, then a * b = b * a, where * represents the group operation. Abelian groups are named after Niels Henrik Abel, a Norwegian mathematician who made significant contributions to the study of groups.
| 3. Can you provide an example of a group that is not Abelian? | |
Ans. Yes, a classic example of a group that is not Abelian is the group of 2x2 matrices with real entries under matrix multiplication. In this group, the order of matrix multiplication matters, and it is not commutative. For example, let A and B be two matrices in this group. In general, AB ≠ BA, which violates the commutative property required for an Abelian group.
| 4. What are some applications of group theory in real-life? | |
Ans. Group theory has various applications in different fields. In physics, it is used to study fundamental particles and their interactions, as well as symmetries in physical systems. In cryptography, group theory plays a crucial role in designing secure encryption algorithms. Group theory is also applied in chemistry to understand molecular symmetry and bonding. Additionally, it has applications in computer science, economics, and even music theory.
| 5. Are all Abelian groups of finite order? | |
Ans. No, not all Abelian groups are of finite order. While Abelian groups can be finite, such as the group of integers modulo n under addition, they can also be infinite. An example of an infinite Abelian group is the group of real numbers under addition. This group satisfies the commutative property, making it an Abelian group, and is infinite in size as the real numbers are uncountable.
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Nov 22, 2024 Last updated |
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