GATE CS 2021 Set 1 — Question 44
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumGroups, Rings & FieldsSets & CombinatoricsEngineering Mathematics
Engineering Mathematics → Sets & Combinatorics → Groups, Rings & Fields
Last updated
Question
Let be a group of order 6, and be a subgroup of such that . Which one of the following options is correct?
Correct answer
(B) G may not be cyclic, but H is always cyclic.
Solution
To determine the correct option, we analyze the properties of the group and its subgroup based on their orders.
A well-known theorem in group theory states that every group of prime order is cyclic. Therefore, must be cyclic.
1. Analysis of the Subgroup
We are given that is a subgroup of and its order satisfies:By Lagrange's Theorem, the order of a subgroup must divide the order of the group. Since , the possible values for are the divisors of strictly between and . Thus:Since both and are prime numbers, the order of is prime.A well-known theorem in group theory states that every group of prime order is cyclic. Therefore, must be cyclic.
2. Analysis of the Group
The group has order . Up to isomorphism, there are exactly two groups of order 6:1.The cyclic group (which is abelian and cyclic).
2.The symmetric group (which is non-abelian and non-cyclic).
Since can be isomorphic to , is not always cyclic (i.e., may not be cyclic).Conclusion
- may not be cyclic (e.g., when ).
- is always cyclic because its order is a prime number ( or ).
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE CS question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Sets & Combinatorics
2026 Set 2 Q3A day can only be cloudy or sunny. The probability of a day being cloudy is , independent of…2026 Set 2 Q5‘When it is raining, peacocks dance.’ Based only on this sentence, which one of the following…2026 Set 2 Q8Figures (i) and (ii) represent intercity highway systems. The black dots represent cities and the…2026 Set 1 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…2026 Set 2 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…