Question

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 !

Answer 1

In every finite $p$-group $G$ of order $p^n$ there is a normal subgroup $N_k$ of order $p^k$ for any $k\le n$. The proof is by induction on $k$. For $k=1$ take a cyclic subgroup of order $p$ in the center, for the step, take a cyclic central subgroup in $G/N_k$ and its preimage in $G$. So if you do not assume that $A\cap B=\{1\}$, one can't say anything specific about the orders. If you assume that $A\cap B=\{1\}$, i.e. your product is semi-direct, then some information (not much, though) can be deduced.

If $|A|=p$ and $A$ is normal, then $A$ is central because in a nilpotent group every non-trivial normal subgroup intersects the center non-trivially. In that case either $B=G$ or $A\cap B=\{1\}$ and $G=A\times B$.

If $|A|=p^k, k\ge 2$, then again you can take a central cyclic of order $p$ subgroup $T$ of $A$, and you will have $G/T=(A/T)(BT/T)$. This reduces your problem to the same problem for a smaller group $G/T$.