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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?

- Every group of order 11 is cyclic.
- Every group of order 111 is cyclic.
- Every group of order 1111 is cyclic.

**Discussion:**

- 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. - Every group of order 111 is cyclic. This is
**true**. . This is a favorite problem for any college level test maker. Order of this group is of the form where p and q are primes. This group is isomorphic to that is the direct product of two cyclic groups of prime order. We have this theorem:The proof of this theorem directly follows from Sylow's 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.** - Every group of order 1111 is cyclic. This is
**true**. 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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[…] Discussion […]

But Zp x Zq is not cyclic. So 2 will be false.

We will ask the author to look into this

Let n be a positive integer. Then the cyclic group C(n) of order n is the only group if and only if (n, phi(n))=1, where phi is the Euler function.

Since(111, phi(111))=(111,72)=3, there are non cyclic group of order 111. Your teacher are so bad.

Thanks for pointing out. This post has been red flagged before.