1. ## Abstract Algebra help

Hello,
I need help with the following problem:

Let G be a finite group, H and N are subgroups of G such that N is normal, and |N| and [G:N] are relatively prime. Show that if H is contained in N, then |H| divides |N|.

I would really help anyone's help on this. Thank you in advance

2. Originally Posted by anlys
Hello,
I need help with the following problem:

Let G be a finite group, H and N are subgroups of G such that N is normal, and |N| and [G:N] are relatively prime. Show that if H is contained in N, then |H| divides |N|.

I would really help anyone's help on this. Thank you in advance
This seems like an easy application of Lagrange's Theorem to me; the order of a subgroup of will always divide the order the the group. Here, if you forget the blurb about N normal and stuff and just look at the last sentence, then N is the group and H is the subgroup, so...

3. Originally Posted by Swlabr
This seems like an easy application of Lagrange's Theorem to me; the order of a subgroup of will always divide the order the the group. Here, if you forget the blurb about N normal and stuff and just look at the last sentence, then N is the group and H is the subgroup, so...
Hello Swlabr,
Thank you for your response. In regards to your comment, I was thinking about using the Lagrange's theorem too. So if we assume that H is contained in N, we would have to show that H is a subgroup of N. If this is true, then we can conclude that |H| divides |N|, right? But, I already tried to prove that H is a subgroup of N, and I am kinda stuck now.

4. Originally Posted by anlys
Hello Swlabr,
Thank you for your response. In regards to your comment, I was thinking about using the Lagrange's theorem too. So if we assume that H is contained in N, we would have to show that H is a subgroup of N. If this is true, then we can conclude that |H| divides |N|, right? But, I already tried to prove that H is a subgroup of N, and I am kinda stuck now.
H is a subgroup of N because it is also a subgroup of G and because it is a subset of N. You need to use both of these facts.

What have you done to prove it so far?

5. Originally Posted by Swlabr
H is a subgroup of N because it is also a subgroup of G and because it is a subset of N. You need to use both of these facts.

What have you done to prove it so far?
Oh, yeah, you're right. It is quite straightforward! I didn't use both of the facts before. Thank you so much for your help!