Non-abelian groups are pervasive in mathematics and physics. 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.

It's often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations $a^4=1$, $b^2=1$ and $bab^ {-1}=a^3$.

We want give some examples of genuinely nonabelian groups. The next example should already be familiar from linear algebra class (where F is usually taken to be R or maybe C).

It relies on a classification result that states that every Hamiltonian group is a direct product of the quaternion group of order 8, an elemetary abelian 2-group, and a periodic abelian group of odd order.

A group is non-abelian or non-commutative if it is not abelian, in other words if not all elements of the group commute. See also abelian groups.

1 Permutations of a group has a large efect on what the group an look like. F he order of G is the cardinality (or n mber of elements) of G. In other words, the order of G is |G s the general linear …

4 days ago · 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 …