2nd October 2019

11

# What is the symmetric group?

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.

Also question is, is d3 an Abelian group?

A non-Abelian group, also sometimes known as a noncommutative group, is a group some of whose elements do not commute. The simplest non-Abelian group is the dihedral group D3, which is of group order six.

Is a3 cyclic?

For example A3 is a normal subgroup of S3, and A3 is cyclic (hence abelian), and the quotient group S3/A3 is of order 2 so it's cyclic (hence abelian), and hence S3 is built (in a slightly strange way) from two cyclic groups. More generally: Definition 1.

What is the alternating group?

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt(n).