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. \(111 = 3 \times 37 \). 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.
- 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