Non-abelian Groups of Order 6

To find the total number of non-abelian groups of order 6, we need to consider all the possible groups of this order and determine which ones are non-abelian.

Possible Groups of Order 6

There are several possible groups of order 6. Let's consider each of them and determine whether they are abelian or non-abelian.

1. Trivial Group: The trivial group is the group containing only the identity element. It is abelian.

2. Cyclic Group of Order 6: The cyclic group of order 6, denoted by C6, is generated by a single element and follows the operation of addition modulo 6. It is abelian.

3. Direct Product of Cyclic Groups: The direct product of two cyclic groups of order 2 and 3, denoted by C2 x C3, contains elements (0,0), (0,1), (1,0), (1,1), (0,2), (1,2) and follows the operation of component-wise addition modulo 2 and 3. It is abelian.

4. Symmetric Group of 3 Elements: The symmetric group of 3 elements, denoted by S3, consists of all possible permutations of three elements. It is non-abelian.

5. Dihedral Group of Order 6: The dihedral group of order 6, denoted by D6, consists of all possible rotations and reflections of a regular hexagon. It is non-abelian.

Conclusion

Out of the possible groups of order 6, only the symmetric group of 3 elements (S3) and the dihedral group of order 6 (D6) are non-abelian. Therefore, the total number of non-abelian groups of order 6 is 1.

Summary:

- Trivial Group: Abelian
- Cyclic Group of Order 6: Abelian
- Direct Product of Cyclic Groups: Abelian
- Symmetric Group of 3 Elements (S3): Non-abelian
- Dihedral Group of Order 6 (D6): Non-abelian

Hence, there is only 1 non-abelian group of order 6.