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Smalls, H; be two distinct subgroups of a finite 3 group G, each of order 2. Let Hebe the smallest subgroup containing H, and H₂. Then the order of H is
(a) always 2
(d) none of the above
(b) always 4
(c) always 8?
(a) always 2
(d) none of the above
(b) always 4
(c) always 8?
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Smalls, H; be two distinct subgroups of a finite 3 group G, each of or...
Understanding the Subgroups of a Finite 3-Group
When analyzing the situation with two distinct subgroups H1 and H2 of a finite 3-group G, each of order 2, we can derive some conclusions based on group theory principles.
Properties of Subgroups
- In a group, the order of a subgroup divides the order of the group (Lagrange's theorem).
- The elements of a subgroup must be closed under the group operation.
Combination of Subgroups
- The smallest subgroup containing both H1 and H2 is denoted as H.
- Since both H1 and H2 are of order 2, each subgroup contains an identity element and one other element.
Order of the Combined Subgroup H
- The combined subgroup H must include the identity element and the elements of both H1 and H2.
- Since H1 and H2 are distinct, they share no common non-identity elements.
- Thus, the combined subgroup H will consist of the identity and the two distinct elements from H1 and H2.
Conclusion on the Order of H
- Therefore, the order of H, which contains the identity and the two distinct elements, is 3.
- However, this contradicts our earlier understanding that subgroups of finite 3-groups can only have orders that are powers of 3.
Final Answer
- Since H must include all distinct elements from both groups, the maximum order H can take is 4 (which includes three distinct elements in addition to the identity).
In conclusion, the order of H is always 4 (option b).
When analyzing the situation with two distinct subgroups H1 and H2 of a finite 3-group G, each of order 2, we can derive some conclusions based on group theory principles.
Properties of Subgroups
- In a group, the order of a subgroup divides the order of the group (Lagrange's theorem).
- The elements of a subgroup must be closed under the group operation.
Combination of Subgroups
- The smallest subgroup containing both H1 and H2 is denoted as H.
- Since both H1 and H2 are of order 2, each subgroup contains an identity element and one other element.
Order of the Combined Subgroup H
- The combined subgroup H must include the identity element and the elements of both H1 and H2.
- Since H1 and H2 are distinct, they share no common non-identity elements.
- Thus, the combined subgroup H will consist of the identity and the two distinct elements from H1 and H2.
Conclusion on the Order of H
- Therefore, the order of H, which contains the identity and the two distinct elements, is 3.
- However, this contradicts our earlier understanding that subgroups of finite 3-groups can only have orders that are powers of 3.
Final Answer
- Since H must include all distinct elements from both groups, the maximum order H can take is 4 (which includes three distinct elements in addition to the identity).
In conclusion, the order of H is always 4 (option b).
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Smalls, H; be two distinct subgroups of a finite 3 group G, each of order 2. Let Hebe the smallest subgroup containing H, and H₂. Then the order of H is(a) always 2(d) none of the above(b) always 4(c) always 8?
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Smalls, H; be two distinct subgroups of a finite 3 group G, each of order 2. Let Hebe the smallest subgroup containing H, and H₂. Then the order of H is(a) always 2(d) none of the above(b) always 4(c) always 8? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Smalls, H; be two distinct subgroups of a finite 3 group G, each of order 2. Let Hebe the smallest subgroup containing H, and H₂. Then the order of H is(a) always 2(d) none of the above(b) always 4(c) always 8? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Smalls, H; be two distinct subgroups of a finite 3 group G, each of order 2. Let Hebe the smallest subgroup containing H, and H₂. Then the order of H is(a) always 2(d) none of the above(b) always 4(c) always 8?.
Smalls, H; be two distinct subgroups of a finite 3 group G, each of order 2. Let Hebe the smallest subgroup containing H, and H₂. Then the order of H is(a) always 2(d) none of the above(b) always 4(c) always 8? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Smalls, H; be two distinct subgroups of a finite 3 group G, each of order 2. Let Hebe the smallest subgroup containing H, and H₂. Then the order of H is(a) always 2(d) none of the above(b) always 4(c) always 8? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Smalls, H; be two distinct subgroups of a finite 3 group G, each of order 2. Let Hebe the smallest subgroup containing H, and H₂. Then the order of H is(a) always 2(d) none of the above(b) always 4(c) always 8?.
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