I was wondering regarding the next question I encountered during my current research:

Given a p-group $G$ of order $ p^n$ that can be decomposed into the product $ G=AB$ of a normal subgroup $A$ and an arbitrary subgroup $ B$ , what can we say about the orders of $A$ and $B$ ? Lagrange's theorem tells us that $A,B$ must be also p-groups, but can we say something about the power of p that their order has?

What if $ B $ was cyclic? Does it change the answer?

If we also know that the rank of $ G $ is $ k $ , will it help us to determine the rank of $ A$ ?

In another direction, given such a decomposition, how unique is it ? i.e.- have we got any way to estimate the number of ways in which we can decompose an arbitrary p-group into such a product?

Hope you'll be able to help. Any reference will be greatfully acknowledged

Thanks !