How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# Cyclic Group problem in NBHM M.Sc. 2013

This a problem from from National Board for Higher Mathematics (NBHM) 2013 based on Cyclic Group Problem for M.Sc Students.

Which of the following statements are true?

1. Every group of order 11 is cyclic.
2. Every group of order 111 is cyclic.
3. Every group of order 1111 is cyclic.

Discussion:

1. Every group of order 11 is cyclic. This is true. We know that (for example from Lagrange's Theorem) that a group of prime order is necessarily cyclic.
2. Every group of order 111 is cyclic. This is true. $111 = 3 \times 37$. This is a favorite problem for any college level test maker. Order of this group is of the form $p \times q$ where p and q are primes. This group is isomorphic to $Z_p \times Z_q$ that is the direct product of two cyclic groups of prime order. We have this theorem:  Let G be a group of order pq, where p,q are prime, p<q, and p does not divide q−1. Then G is cyclic. The proof of this theorem directly follows from Sylow's Theorem.
3. Every group of order 1111 is cyclic. This is true. $1111 = 11 \times 101$ and using the argument of the previous problem we conclude that it is cyclic.

Critical Ideas: Lagrange's Theorem, Sylow's Theorem, Classification of finite abelian groups

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INMO 2012 Problems

INMO 2018 Problem 6
The text passage contains a series of true/false statements about the cyclicity of groups of specific orders, followed by explanations for each statement. The passage does not contain a direct mathematical question but rather a series of assertions and their justifications.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# Cyclic Group problem in NBHM M.Sc. 2013

This a problem from from National Board for Higher Mathematics (NBHM) 2013 based on Cyclic Group Problem for M.Sc Students.

Which of the following statements are true?

1. Every group of order 11 is cyclic.
2. Every group of order 111 is cyclic.
3. Every group of order 1111 is cyclic.

Discussion:

1. Every group of order 11 is cyclic. This is true. We know that (for example from Lagrange's Theorem) that a group of prime order is necessarily cyclic.
2. Every group of order 111 is cyclic. This is true. $111 = 3 \times 37$. This is a favorite problem for any college level test maker. Order of this group is of the form $p \times q$ where p and q are primes. This group is isomorphic to $Z_p \times Z_q$ that is the direct product of two cyclic groups of prime order. We have this theorem:  Let G be a group of order pq, where p,q are prime, p<q, and p does not divide q−1. Then G is cyclic. The proof of this theorem directly follows from Sylow's Theorem.
3. Every group of order 1111 is cyclic. This is true. $1111 = 11 \times 101$ and using the argument of the previous problem we conclude that it is cyclic.

Critical Ideas: Lagrange's Theorem, Sylow's Theorem, Classification of finite abelian groups

### Try a online trial session of Cheenta I.S.I. M.Math, IIT JAM, TIFR Entrance Program. Mail us at helpdesk@cheenta.com

INMO 2012 Problems

INMO 2018 Problem 6
user
The text passage contains a series of true/false statements about the cyclicity of groups of specific orders, followed by explanations for each statement. The passage does not contain a direct mathematical question but rather a series of assertions and their justifications.

TYPE: NO_QUESTION