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No of unique normal subgroup of order 4 in D8?
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No of unique normal subgroup of order 4 in D8?
Number of Unique Normal Subgroups of Order 4 in D8
D8 refers to the dihedral group of order 8, which consists of 8 elements. In order to determine the number of unique normal subgroups of order 4 in D8, we need to understand the structure of D8 and its subgroups.
Dihedral Group D8:
The dihedral group D8 is a group of symmetries of a regular octagon. It can be generated by two elements, a rotation of 45 degrees and a reflection. Let's denote the rotation by R and the reflection by F. The elements of D8 can be represented as {R^0, R^1, R^2, R^3, F, FR, FR^2, FR^3}, where R^0 represents the identity element.
Subgroups of D8:
D8 has several subgroups of different orders. Some of the important subgroups are:
1. Trivial Subgroup: The subgroup {R^0} consists only of the identity element.
2. Order 2 Subgroups: There are two order 2 subgroups in D8, {R^0, F} and {R^0, FR^2}.
3. Order 4 Subgroups: There are three order 4 subgroups in D8, namely {R^0, R^2, F, FR^2}, {R^0, R^2, FR, FR^3}, and {R^0, R^2, FR^3, FR}.
4. Order 8 Subgroup: The entire group D8 itself is an order 8 subgroup.
Normal Subgroups:
A subgroup H of a group G is said to be normal if and only if for every g in G, gH = Hg, where gH represents the left coset of H in G and Hg represents the right coset of H in G.
Analysis of Order 4 Subgroups:
Now, let's analyze the order 4 subgroups of D8 to determine which ones are normal.
1. {R^0, R^2, F, FR^2}: This subgroup is not normal in D8. We can see this by considering the element FR^3 in D8. If we multiply FR^3 with any element of this subgroup, the result will not be in the subgroup. Hence, it is not normal.
2. {R^0, R^2, FR, FR^3}: This subgroup is also not normal in D8. A similar argument as above can be used to show that it is not normal.
3. {R^0, R^2, FR^3, FR}: This subgroup is normal in D8. We can verify this by multiplying any element of this subgroup with FR^3 or FR.
Therefore, there is only one unique normal subgroup of order 4 in D8, which is {R^0, R^2, FR^3, FR}.
Conclusion:
The number of unique normal subgroups of order 4 in D8 is 1, which is
D8 refers to the dihedral group of order 8, which consists of 8 elements. In order to determine the number of unique normal subgroups of order 4 in D8, we need to understand the structure of D8 and its subgroups.
Dihedral Group D8:
The dihedral group D8 is a group of symmetries of a regular octagon. It can be generated by two elements, a rotation of 45 degrees and a reflection. Let's denote the rotation by R and the reflection by F. The elements of D8 can be represented as {R^0, R^1, R^2, R^3, F, FR, FR^2, FR^3}, where R^0 represents the identity element.
Subgroups of D8:
D8 has several subgroups of different orders. Some of the important subgroups are:
1. Trivial Subgroup: The subgroup {R^0} consists only of the identity element.
2. Order 2 Subgroups: There are two order 2 subgroups in D8, {R^0, F} and {R^0, FR^2}.
3. Order 4 Subgroups: There are three order 4 subgroups in D8, namely {R^0, R^2, F, FR^2}, {R^0, R^2, FR, FR^3}, and {R^0, R^2, FR^3, FR}.
4. Order 8 Subgroup: The entire group D8 itself is an order 8 subgroup.
Normal Subgroups:
A subgroup H of a group G is said to be normal if and only if for every g in G, gH = Hg, where gH represents the left coset of H in G and Hg represents the right coset of H in G.
Analysis of Order 4 Subgroups:
Now, let's analyze the order 4 subgroups of D8 to determine which ones are normal.
1. {R^0, R^2, F, FR^2}: This subgroup is not normal in D8. We can see this by considering the element FR^3 in D8. If we multiply FR^3 with any element of this subgroup, the result will not be in the subgroup. Hence, it is not normal.
2. {R^0, R^2, FR, FR^3}: This subgroup is also not normal in D8. A similar argument as above can be used to show that it is not normal.
3. {R^0, R^2, FR^3, FR}: This subgroup is normal in D8. We can verify this by multiplying any element of this subgroup with FR^3 or FR.
Therefore, there is only one unique normal subgroup of order 4 in D8, which is {R^0, R^2, FR^3, FR}.
Conclusion:
The number of unique normal subgroups of order 4 in D8 is 1, which is
Community Answer
No of unique normal subgroup of order 4 in D8?
Number of Unique Normal Subgroups of Order 4 in D8
D8 refers to the dihedral group of order 8. It is a group of symmetries of a regular octagon. The dihedral group D8 has 16 elements, including 8 rotations and 8 reflections. In this response, we will determine the number of unique normal subgroups of order 4 in D8.
Dihedral Group D8
The dihedral group D8 can be represented by the set of generators {r, s}, where r represents a rotation and s represents a reflection. The generators satisfy the following relations:
r^8 = e (identity element)
s^2 = e
rs = sr^(-1)
Properties of Normal Subgroups
Before we find the normal subgroups of order 4, it is important to understand the properties of normal subgroups:
1. Normal subgroups are invariant under conjugation. This means that for any element g in the group, gHg^(-1) = H, where H is a normal subgroup.
2. Normal subgroups are a union of conjugacy classes.
Determining the Normal Subgroups of Order 4
To find the normal subgroups of order 4 in D8, we need to consider the elements of order 4 in D8. The possible elements of order 4 are rotations of 90 degrees.
1. Identity Subgroup: The subgroup {e} consisting of only the identity element is a normal subgroup of order 1.
2. Subgroup Generated by a Rotation: The subgroup generated by a rotation of order 4, denoted as, is a normal subgroup of order 4. This subgroup is invariant under conjugation as any rotation conjugated with another element will still result in a rotation.
3. Subgroup Generated by a Reflection: The subgroup generated by a reflection of order 2, denoted as, is also a normal subgroup of order 2. This subgroup is invariant under conjugation as any reflection conjugated with another element will still result in a reflection.
4. Subgroup Generated by a Reflection and a Rotation: The subgroup generated by a reflection and a rotation, denoted as, is another normal subgroup of order 4. This subgroup is also invariant under conjugation.
5. Trivial Subgroup: Finally, the whole group D8 itself is a normal subgroup of order 8.
Therefore, the number of unique normal subgroups of order 4 in D8 is 4. These subgroups are the identity subgroup, the subgroup, the subgroup , and the subgroup .
Summary
In summary, the dihedral group D8 has a total of 4 unique normal subgroups of order 4. These subgroups are the identity subgroup, the subgroup generated by a rotation of order 4, the subgroup generated by a reflection of order 2, and the subgroup generated by a reflection and a rotation. These subgroups are invariant under conjugation and satisfy the properties of normal subgroups.
D8 refers to the dihedral group of order 8. It is a group of symmetries of a regular octagon. The dihedral group D8 has 16 elements, including 8 rotations and 8 reflections. In this response, we will determine the number of unique normal subgroups of order 4 in D8.
Dihedral Group D8
The dihedral group D8 can be represented by the set of generators {r, s}, where r represents a rotation and s represents a reflection. The generators satisfy the following relations:
r^8 = e (identity element)
s^2 = e
rs = sr^(-1)
Properties of Normal Subgroups
Before we find the normal subgroups of order 4, it is important to understand the properties of normal subgroups:
1. Normal subgroups are invariant under conjugation. This means that for any element g in the group, gHg^(-1) = H, where H is a normal subgroup.
2. Normal subgroups are a union of conjugacy classes.
Determining the Normal Subgroups of Order 4
To find the normal subgroups of order 4 in D8, we need to consider the elements of order 4 in D8. The possible elements of order 4 are rotations of 90 degrees.
1. Identity Subgroup: The subgroup {e} consisting of only the identity element is a normal subgroup of order 1.
2. Subgroup Generated by a Rotation: The subgroup generated by a rotation of order 4, denoted as
3. Subgroup Generated by a Reflection: The subgroup generated by a reflection of order 2, denoted as
4. Subgroup Generated by a Reflection and a Rotation: The subgroup generated by a reflection and a rotation, denoted as
5. Trivial Subgroup: Finally, the whole group D8 itself is a normal subgroup of order 8.
Therefore, the number of unique normal subgroups of order 4 in D8 is 4. These subgroups are the identity subgroup, the subgroup
Summary
In summary, the dihedral group D8 has a total of 4 unique normal subgroups of order 4. These subgroups are the identity subgroup, the subgroup generated by a rotation of order 4, the subgroup generated by a reflection of order 2, and the subgroup generated by a reflection and a rotation. These subgroups are invariant under conjugation and satisfy the properties of normal subgroups.
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No of unique normal subgroup of order 4 in D8?
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No of unique normal subgroup of order 4 in D8? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about No of unique normal subgroup of order 4 in D8? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for No of unique normal subgroup of order 4 in D8?.
No of unique normal subgroup of order 4 in D8? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about No of unique normal subgroup of order 4 in D8? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for No of unique normal subgroup of order 4 in D8?.
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