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Concept of Normal Subgroup Video Lecture | Mathematics for Competitive Exams
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FAQs on Concept of Normal Subgroup Video Lecture - Mathematics for Competitive Exams
| 1. What is a normal subgroup? |  |
Ans. A normal subgroup is a subgroup of a group that is invariant under conjugation by any element of the larger group. In other words, if N is a normal subgroup of a group G, then for every element g in G and every element n in N, the conjugate of n by g (i.e., gng⁻¹) is also in N.
| 2. How is a normal subgroup different from a regular subgroup? | |
Ans. A normal subgroup differs from a regular subgroup in the sense that it remains invariant under conjugation by any element of the larger group, while a regular subgroup may not have this property. In other words, if N is a normal subgroup of a group G, then for every element g in G and every element n in N, the conjugate of n by g (i.e., gng⁻¹) is also in N.
| 3. What is the significance of normal subgroups in group theory? | |
Ans. Normal subgroups have significant importance in group theory as they provide a way to study and analyze the structure of groups. They allow us to define quotient groups, which are formed by partitioning the group elements based on the equivalence relation induced by the normal subgroup. This concept helps in understanding the properties and relations between different groups.
| 4. How can we determine if a subgroup is normal? | |
Ans. To determine if a subgroup is normal, we can apply the criterion known as the normal subgroup test. According to this test, a subgroup H of a group G is normal if and only if for every element g in G, the conjugate of H by g (i.e., gHg⁻¹) is a subset of H. If this condition holds true for every element of G, then H is a normal subgroup.
| 5. Can a group have multiple normal subgroups? | |
Ans. Yes, a group can have multiple normal subgroups. In fact, every group has at least two normal subgroups: the trivial subgroup {e} and the entire group itself. Additionally, there can be non-trivial normal subgroups that lie between these extremes. These normal subgroups play a crucial role in group theory and help in analyzing the structure and properties of the group.
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Dec 23, 2024 Last updated |
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