🛈 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

Examples

1. 
2. 
3. 
4. 
5. 
6. 
7. $ latex (\Bbb C- \{0\},.)$, where refers to multiplication of complex numbers.
8. 
9. Integers modulo n,
10. , the symmetric group.
11. Any group of order < 6
12. Any group of order where is prime.
13. Any cyclic group is abelian.
14. Any group of prime order is cyclic and hence abelian.

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Counter-Examples (or Non-Abelian Groups)

1. Symmetric groups
2. 
3. , the group of Quaternions.
4. 
5. 
6. , where .