suppose xH = yH for two elements x,y of G.
then y-1x is in H. but H ⊆ J! so y-1x is in J.
thus xJ = yJ.
perhaps an example will make this clearer:
we'll use an abelian group G, so we don't have to worry about normality.
let G = Z12 under addition mod 12.
let J = {0,2,4,6,8,10}, and let H = {0,4,8}.
clearly J contains H.
the cosets x+J:
J = {0,2,4,6,8,10}
1+J = {1,3,5,7,9,11}
(these cosets have "other names", for example 4+J = J, and 7+J = 1+J).
the cosets x+H:
H = {0,4,8}
1+J = {1,5,9}
2+J = {2,6,10}
3+J = {3,7,11}
notice anything?
when H partitions G, it "respects the partition induced by J", we just chop the cosets by J into SMALLER cosets by H. so:
J = H U (2+H)
1+J = (1+H) U (3+H)
so any two elements in the same coset of H are in the same coset of J (we get cosets of J by "lumping together cosets of H").
specifically the map x+H --> x+J takes:
H-->J
2+H-->J (and 2 is in J, this works)
1+H-->1+J
3+H-->1+J (and 3 is in 1+J, so this is fine, as well).
you can also look at it this way:
the cosets xJ chop G up into "J sized pieces"(even if J isn't normal).
since H is a subgroup of J, we can, in turn, chop J into "H sized pieces"
and use this to chop the J-pieces xJ into H-pieces x(yH).
since J is BIGGER, G/J is "chunkier" (bigger pieces), while G/H is "finer" (smaller pieces),
and each bigger chunk of G/J is composed of smaller chunks of G/H.
if both subgroups are normal, then we have a group structure on G/J and G/H
and there is a nice relationship between G/J and G/H, the same relationship enjoyed by J and H (the cosets just "magnify it" by a factor of the indices of the respective subgroups involved).
