UPSC Exam > UPSC Questions > Smalls, H; be two distinct subgroups of a fin...
Start Learning for Free
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?
Most Upvoted Answer
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).
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
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?
Question Description
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?.
Solutions 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? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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 detailed solution 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup