Examples and Counter Examples for Abelian Groups

Note: This is the first of “Examples and Counter Examples” series. I plan on expanding such posts in course of time, as I come across examples.

Interesting fact: These are the OEIS lists of number of distinct groups of order n: number of groups,number of abelian groups, number of non-abelian groups


  1. \Bbb Z
  2. \Bbb Q
  3. \Bbb R
  4. Integers modulo n \Bbb Z / n \Bbb Z
  5. S_n,n\le2, the symmetric group.
  6. A_n \subset S_n,\forall n\ge1
  7. Any group of order < 4

Counter-Examples (or Non-Abelian Groups)

  1. Symmetric groups S_n,n\ge3