Wow, this forum is amazing! I didn't realize places like this exist and I'm really looking forward to getting stuck in to helping people. Anyway, my first post is unfortunately pleading for help with a subject that I have very little in common with. However, I have an upcoming assessment and am really struggling. Any help that you could give me would be very much appreciated
1/ Let G be a finite group and N be a normal subgroup of G such that gcd(|N|,|G/N|)=1. Show that N is the only subgroup of G with order |N|.
I have the feeling that this has something to do with the Second Isomorphism Theorem, though I am not sure. I have tried everything I can think of involving it but to no avail
2/ Give a decomposition series for the dihedral group of order 28. To which well-known groups are the composition factors isomorphic? I'm lost on this one
Thank you in advance for any help that you can give
We can prove something stronger.
Theorem: Let be a finite group and a normal subgroup with and if is a subgroup with then .
Proof: Form the factor group . Let and consider the coset , we will show that proving that and completing the proof. Note, it is sufficient to prove that the order of in the group is . Let be the order of . Let be the order of in the group . Now thus it means . But by Lagrange's theorem and divides so divides by Lagrange's theorem, thus divides . But then because they are relatively prime.
The dihedral group can be thought of .
Now is a composition series. Because which is simple. And which is simple. And which is simple.
Note, all the composition factor groups are abelian which means this dihedral group is solvable. And furthermore, any composition series must has basically the composition series given above by the Jordan-Holder theorem (one of my favorite theorems from all math).