Thread: Factor groups for the quaternion group?

I know the subgoups of the quaternion group (and that they are all normal), but i'm not sure how to determine the factor group Q8/N for each of the normal subgroups N.

Thanks for your help.

Originally Posted by Louise

I know the subgoups of the quaternion group (and that they are all normal), but i'm not sure how to determine the factor group Q8/N for each of the normal subgroups N.

Thanks for your help.

if or , then or respectively. the only (normal) subgroup of order 2 is then (here is the cyclic group of order 2.)

you probably know that is also called the Klein 4-group.

How do you do the calculation Q8/{1,-1} to give {1,i,j,k}?

Originally Posted by Louise

How do you do the calculation Q8/{1,-1} to give {1,i,j,k}?

Q_8/{1,-1} is NOT {1,i,j,k} but the images of these elements in the factor group, just Noncommalg denoted by putting a bar over the elements i,j,k.

Why is this thing isomorphic with Z_2 x Z_2? Well, check that any of the elements 1, i, j, k squared gives you an element in {1,-1}...

Tonio