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: 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.

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

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.

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.