Let Q be the set of eight elements 1 i j k We define the bin

Let Q be the set of eight elements, {±1, ±i, ±j, ±k}. We define the binary operationsbytherulesij=ji=k,jk=kj=i,ki=ik=j,i^2 =j^2 =k^2 = 1, 1i = i, and (1)^2 = 1. The rules with i, j, and k can be illustrated by using the diagram below.

Solution

The sub gruops of Q are A = { 1 , -1 , i ,-i } A is abelian and cyclic A =< i> , = < - i > both i , -i are the generators

similarly the other subgroups are B = { 1 ,-1 , j ,-j } is abelian and cyclic with generators j , - j

and C = { 1 , -1 - k , -k} is abelian and cyclic with generators k and - k