# Thread: Normal Subgroup with Prime Index

1. ## Normal Subgroup with Prime Index

Hi, my problem-problem goes like this:

Prove that if H is a normal subgroup of G with prime index p, then for all subgroups K of G

i)K is a subgroup of G, or
ii)G=HK

I have a feeling that an isomorphism theorem is in order, but every arrangement I've tried fails...or else I'm just not seeing it. Any suggestions would be warmly appreciated. Also, in the event that I ever come back with another question, how should I get started with Latex?

2. Originally Posted by mstrfrdmx
Hi, my problem-problem goes like this:

Prove that if H is a normal subgroup of G with prime index p, then for all subgroups K of G

i)K is a subgroup of G, or
ii)G=HK

I have a feeling that an isomorphism theorem is in order, but every arrangement I've tried fails...or else I'm just not seeing it. Any suggestions would be warmly appreciated. Also, in the event that I ever come back with another question, how should I get started with Latex?

Write back stating correctly the question: number (1) is nonsensical in view of what you wrote just one line above it.

Tonio

3. eh, sorry:

Prove that if H is a normal subgroup of G with prime index p, then for all subgroups K of G

i)K is a subgroup of H, or
ii)G=HK

4. Originally Posted by mstrfrdmx
eh, sorry:

Prove that if H is a normal subgroup of G with prime index p, then for all subgroups K of G

i)K is a subgroup of H, or
ii)G=HK

Now we're talking. Check and prove the following hints:

1) If $\displaystyle K\lneq H\lneq G\,,\,\,G$ a group, then $\displaystyle [G:H]\leq[G:K]$ , with strict inequality if the indexes are finite ;

2) If $\displaystyle K\nleq H$ , then $\displaystyle H\varsubsetneqq HK$

Tonio