Question

Prove that quaternion group under multiplication is non abelian

Answer 1

Answer:

Since, all 4 properties of group are satisfied G is a group. Thus, G is commutative. A commutative group is called as an abelian group. Thus, cube roots of unity form a finite abelian group under multiplication.

Answer 2

Note :

Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Cyclic group : A group G is called a cyclic group , if there exists an element a ∈ G , such that every element x ∈ G can be written as x = aⁿ for some integer n . And the element a is called the generator of G .

Solution :

Please refer to the attachments .