# 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

### Examples

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.

### 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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