# Order of the factor group G/N

• Nov 29th 2007, 07:25 PM
temp31415
Order of the factor group G/N
Question: Suppose G is a group with |G| = 30 and N is a normal subgroup of G. WHat are the possible orders for the group G/N

I know by Lagrange's theorem that |N| divides |G| so |N| = {1,2,3,5,10,15,30}.

I also know that |G/N| / |N| = |G| / |N| so I believe |G/N| = |N| = {1,2,3,5,10,15,30}

Am I off base or is there a better way to approach this?

Thanks
• Nov 29th 2007, 07:35 PM
ThePerfectHacker
Quote:

Originally Posted by temp31415
Question: Suppose G is a group with |G| = 30 and N is a normal subgroup of G. WHat are the possible orders for the group G/N

I know by Lagrange's theorem that |N| divides |G| so |N| = {1,2,3,5,10,15,30}.

I also know that |G/N| / |N| = |G| / |N| so I believe |G/N| = |N| = {1,2,3,5,10,15,30}

Am I off base or is there a better way to approach this?

Thanks

Thus, the possible normal subgroups that |N| can be are: 1,2,3,5,6,10,15,30
Thus, |G/N|=|G|/|N| are: 1,2,3,5,6,10,15,30

Just note this might not be the best answer. Meaning using some additional results we might be able to disregard which orders are possible and impossible. But that is a more difficult task.