## Is sn a non-Abelian?

Since these elements belong to Sn for n ≥ 3, it follows that Sn is non- abelian (for n ≥ 3).

### How can I prove my S3 is non-Abelian?

S3 is not abelian, since, for instance, (12) · (13) = (13) · (12). On the other hand, Z6 is abelian (all cyclic groups are abelian.) Thus, S3 ∼ = Z6.

#### What is the order of Sn?

2 . the symmetric group on S will be denoted by Sn. The number of elements of Sn is found in the following theorem. Theorem 6.2 The order of Sn is n!, where 0!

**What is an example of a non-abelian group?**

One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory.

**Is abelian a S5?**

The symmetric group S5 is defined to be the group of all permutations on a set of five elements, ie, the symmetric group of degree five. In particular, it is a symmetric group of prime degree and it is denoted by S5. A group generated by a single element is called cyclic and we know that cyclic groups are abelian.

## Is A3 abelian?

a) The group of even permutations A3 has three elements, hence it is abelian. The quotient S3/A3 has two elements and therefore it is also abelian. Thus S3 is metabelian.