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.
- 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