# g or n

1. ### Let N be subgroup of the center of G. Show if G/N is a cyclic group, then G abelian

Let N be a subgroup of the center of G. Show that if G/N is a cyclic group, then G must be abelian. suppose G/N is cyclic. this means G/N is generated by a single coset xN, for some x in G. so every element of G is of the form x^kn, for some integer k, and some n in N, and since N is...
2. ### G/N homomorphic image

If G is an abelian group and N is a subgroup of G, show that G/N is an abelian group observing that G/N is a homomorphic image of G. Please show steps, I'm super confused. Thanks!
3. ### G/N

If G is the group of all nonzero real numbers under multiplication and N is the subgroup of all positive real numbers, write out G/N by exhibiting the cosets of N in G, and construct the multiplication in G/N.
4. ### 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}...