If there is a group G of order pqr
(for distict primes p, q and r)
with subgroups H and K of G,
where the order of H is pq
and the order of K is qr
I am trying to show that the order of must be q.
Not sure where to run with this one?
I thought maybe that an intersection of two subgroups may be a subgroup of either of those subgroups,
ie, is a subgroup of H, so the order of must divide the order of H,
and also is a subgroup of K,
so the order of must divide the order of K
but I am not sure if that is valid or not (is the intersection of two subgroups (H and K) a subgroup itself of either of the two orginal subgroups (H or K)?
Any help appreciated!