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Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is
(a) always 2
(b) always 4
(c) always 8
TIFR-2014
(d) none of the above?
(a) always 2
(b) always 4
(c) always 8
TIFR-2014
(d) none of the above?
Most Upvoted Answer
Let H. H₂ be two distinct subgroups of a finite group G, each of order...
Understanding the Problem
When considering two distinct subgroups H1 and H2 of a finite group G, both of order 2, we need to analyze the structure of the subgroup H generated by these two.
Subgroup Properties
- Order of Subgroups: Each subgroup H1 and H2 has order 2. This means each consists of the identity element and one other element.
- Distinctness: Since H1 and H2 are distinct, they have different non-identity elements. Let's denote these elements as a (from H1) and b (from H2).
Generating the Subgroup H
- Elements in H: The subgroup H generated by H1 and H2 will include:
- The identity element e,
- The element a from H1,
- The element b from H2,
- The product ab (assuming it is not equal to e, a, or b).
Analysis of Order of H
- Closure Property: Since H is a subgroup, it must be closed under the group operation.
- Case of ab: The product ab is critical:
- If ab = e, then the group H contains only the elements e, a, and b, giving it an order of 3.
- If ab ≠ e, then H will include e, a, b, and ab, resulting in an order of 4.
Conclusion
- The order of the subgroup H generated by two distinct subgroups of order 2 can be:
- 3 (if ab = e, which is not possible here since they are distinct)
- 4 (if ab ≠ e).
Thus, the order of H is always 4.
Final Answer
The correct option is (b) always 4.
When considering two distinct subgroups H1 and H2 of a finite group G, both of order 2, we need to analyze the structure of the subgroup H generated by these two.
Subgroup Properties
- Order of Subgroups: Each subgroup H1 and H2 has order 2. This means each consists of the identity element and one other element.
- Distinctness: Since H1 and H2 are distinct, they have different non-identity elements. Let's denote these elements as a (from H1) and b (from H2).
Generating the Subgroup H
- Elements in H: The subgroup H generated by H1 and H2 will include:
- The identity element e,
- The element a from H1,
- The element b from H2,
- The product ab (assuming it is not equal to e, a, or b).
Analysis of Order of H
- Closure Property: Since H is a subgroup, it must be closed under the group operation.
- Case of ab: The product ab is critical:
- If ab = e, then the group H contains only the elements e, a, and b, giving it an order of 3.
- If ab ≠ e, then H will include e, a, b, and ab, resulting in an order of 4.
Conclusion
- The order of the subgroup H generated by two distinct subgroups of order 2 can be:
- 3 (if ab = e, which is not possible here since they are distinct)
- 4 (if ab ≠ e).
Thus, the order of H is always 4.
Final Answer
The correct option is (b) always 4.
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Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above?
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Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above?.
Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above?.
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Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above?, a detailed solution for Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above? has been provided alongside types of Let H. H₂ be two distinct subgroups of a finite group G, each of order 2. Let H be the smallest subgroup containing H_{1} and H_{2} Then the order of H is(a) always 2(b) always 4(c) always 8TIFR-2014(d) none of the above? theory, EduRev gives you an
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