Well, my teacher has given us a study guide with no answers. I guess she is busy....anywho...if anyone could help I would really appreciate it.

Suppose that G is a group and that A is a normal subgroup of G.

1.) Suppose that A≤J≤G. Show that A is normal to J.

2.) Suppose that A≤J≤G. show that J/A ≤ G/A.

Attempt:

1.) Suppose that A≤J≤G (subgroup notation). We also suppose that A is a normal Subgroup of G. This means that for every x in G, xA = Ax. So really, all I have are definitions and assumptions and not sure where to go to from there. I was thinking along that lines that since A is normal to G it would automatically follow that A is normal to J since J is a subgroup of G and A is a subgroup of J.

2.) In order to prove that something is a subgroup of another then we have to show 3 things or two things. I also know that

J/A = { gA | g is in J } and G/A = {gA | g is in G}.....

a.) first we need to show that J/A is nonempty.

b.) x in J/A and y in J/A imply xy is also in J/A.

c.) x in J/A implies that x inverse is also in J/A.

If I could get any help I would really appreciate it.