Suppose A, B are normal abelian subgroups of some finite group G. Let [G:A]=m, [G:B]=n, where gcd(m,n)=1.
Can G be non-abelian?
I've been attempting to show that G must be abelian, but I'm starting to think that this isn't necessarily true.
I can show that |G|=|AB|=mnq, |A∩B|=q where q is some positive integer. I can also show that there are cases in which q does not equal 1 (Let G=Z2xZ3xZ4) (if q=1, AxB would then be isomorphic to AB, and so G would be abelian.).