Quaternion Group:

 Quaternion Group:

The quaternion group 𝑄8 is a non-abelian group of order 8, isomorphic to 

the certain eight elements subset of the quaternions under multiplication. It is given by

𝑄8 = {±1, ±𝑖, ±𝑗, ±𝑘}.

Where 𝑖^2 = 𝑗^2 = 𝑘^2 = −1, and

𝑖.𝑗 = 𝑘 = −𝑗. 𝑖

𝑗. 𝑘 = 𝑖 = −𝑘.𝑗

𝑘. 𝑖 = 𝑗 = −𝑖. 𝑘.

Since 𝑖.𝑗 ≠ 𝑗. 𝑖, therefore it is non-abelian. There are 6 subgroups of 𝑄8 of order 1,2,4 and 8. These are

 𝐻1 = 1 , 𝐻4 = {±1, ±𝑗 }

 𝐻2 ={1, −1} , 𝐻5 ={ ±1, ±𝑘 }

 𝐻3 = {±1, ±𝑖} , 𝐻6 ={ ±1, ±𝑖, ±𝑗, ±𝑘} = 𝑄8

All these subgroups are cyclic and abelian. 

Properties:

a) The quaternion group 𝑄8 has the same order as the dihedral group 

𝐷4 =< 𝑎, 𝑏: 𝑎^4 = 𝑏^2 = (𝑎𝑏)^2 = 1 >.

b) Every subgroup of 𝑄8 is a normal subgroup.

c) The center and the commutator subgroup of 𝑄8 is the subgroup {1, −1}.

d) The factor group 𝑄8/{1,−1}

is isomorphic to the Klien four group 𝐾4.

Q8 is solvable group.

Q8 is nilpotent group

Q8 meta ablian group.