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?
Discussion:
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.