Hello,
Could you help me in solving this problem?
---- Let G group, and H, K subgroups of G. Prove that if [G:H]=h and
[G:K]=k then lcm(h,k)|[G:H K] <= h.k and if H or K are
normal subgroups of G then
[G:H intersection K] | h.k ----
I could prove that [G:H K] <= h.k but i cannot prove
lcm(h,k)|[G:H intersection K] and if H or K are normal subgroups of G
then
[G:H K] | h.k
Pleas i really appreciate very much your help and i'm sorry for my bad english
Cheers,
RP
If are normal subgroups then is a normal subgroup. This means we can form the factor group .
For the second part we wil use Lagrange's theorem. We will embed* in . Then it will follow that the index of in must divide and that would complete the proof. Note that is an embedding (you need to show it is well-defined an a homomorphism).
*)Given a group and to embed in means to identity as an isomorphic subgroup of i.e. find a one-to-one homorphism . Then and is a subgroup of because a homomorphic image of a subgroup remains a subgroup.