Non-Cyclic Abelian Groups

It is well known fact that every cyclic group is abelian. But the converse is not true always. i.e. There exists groups that are abelian but not cyclic.

Uncountable abelian groups

The groups such as (\Bbb R,+) and (\Bbb C, +) are abelian groups. Since any infinite cyclic group is isomorphic to (\Bbb Z,+) and hence is countable, the above mentioned groups cannot be cylic as they are uncountable. In fact any uncountable group is non-cyclic.

Finite abelian groups

The Fundamental theorem of finite abelian groups states that ” Every finite abelian group is isomorphic to direct product of cyclic groups”. Also \Bbb Z_m \times \Bbb Z_n is cyclic if and only if gcd(m,n)=1. Using above facts, we can conclude that any abelian group which is isomorphic to \Bbb Z_m \times \Bbb Z_n with gcd(m,n) \ne 1, is non-cyclic but abelian.

Countable abelian groups

  1. \Bbb Z \times \Bbb Z is abelian but not cyclic. Note that here every element is of infinite order.
  2. (\Bbb Q-\{0\}, .) is abelian but not cyclic. For, if suppose it was cyclic, then (\Bbb Q-\{0\}, .) will be isomorphic to (\Bbb Z, +). But (\Bbb Q-\{0\}, .) contains a non-trivial finite order element whereas (\Bbb Z, +) does not.
  3. \Bbb Q / \Bbb Z is abelian since \Bbb Q is abelian. Every element of this group is of finite order since o(\frac p q + \Bbb Z)=q where gcd(p,q)=1. Also this group is infinite since \Bbb \{\frac 1 n + \Bbb Z: n\in \Bbb N\} is an infinite subset of \Bbb Q / \Bbb Z. Hence \Bbb Q / \Bbb Z is an infinite non-cyclic abelian group.