Examples and Counterexampels: Abelian Groups

🛈 This materials is useful for Unit 2 in CSIR-NET and GATE Mathematical Sciences.

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 Q-\{0\}, .)
  4. (\Bbb R,+)
  5. (\Bbb R-\{0\}, .)
  6. (\Bbb C,+)
  7. $ latex (\Bbb C- \{0\},.)$, where . refers to multiplication of complex numbers.
  8. (M_n(\Bbb R),+)
  9. Integers modulo n, \Bbb Z / n \Bbb Z
  10. S_n,n\le2, the symmetric group.
  11. Any group of order < 6
  12. Any group of order p^2 where p is prime.
  13. Any cyclic group is abelian.
  14. Any group of prime order is cyclic and hence abelian.

See comments for link to related articles in the blog.

Counter-Examples (or Non-Abelian Groups)

  1. Symmetric groups S_n,n\ge3
  2. A_n \subset S_n,\forall n\ge4
  3. Q_8, the group of Quaternions.
  4. (GL_n(\Bbb R), .), n>1
  5. (SL_n(\Bbb R), .), n>1
  6. (D_n,\cdot), n>2, where o(D_n)=2n.

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