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Show that a finite non-abelian $p$-group cannot split over its center.

I'd be happy for some clues.

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    @Aaron: Splitting over a central subgroup is always a direct product. In general it is only semi.2012-07-15

2 Answers 2

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This definition is not very common, so it may be worth mentioning here:

Definition. Let $G$ be a group. A subgroup $K$ of $G$ is said to be co-central in $G$ if $G=Z(G)K$.

Theorem. Let $G$ be a group. If $K$ is co-central in $G$, then $Z(K)\subseteq Z(G)$.

Proof. Let $z\in Z(K)$, and $x\in G$. We need to show that $zx=xz$. Since $G=Z(G)K$, there exists $a\in Z(G)$ and $k\in K$ such that $x=ak$. Then $\begin{align*} zx &= zak\\ &= azk &&\text{(since }a\in Z(G)\text{)}\\ &= akz &&\text{(since }z\in Z(K)\text{ and }k\in K\text{)}\\ &= xz. \end{align*}$ Thus, $z\in Z(G)$, as claimed. $\Box$

Now assume that we can write a $p$-group $P$ as $P=Z(P)H$ with $H$ a subgroup. Then by the lemma, $Z(H)\subseteq Z(P)$. If $P$ were split over $Z(P)$, then we would have $Z(P)\cap H = \{e\}$, hence $Z(H)=\{e\}$.

Why is that a very big problem for $H$, which can only be solved if $H=\{e\}$?

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Suppose $P=Z(P)H$ with $1 nontrivial and $H\cap Z(P)=1$. Two questions:

  • Can $Z(H)\subseteq H$ and $Z(P)$ intersect nontrivially?
  • Can $Z(H)=1$? Observe that $H$ is a $p$-group.

Show that every element of $Z(P)Z(H)$ commutes with every element of $Z(P)H=P$, and therefore derive the containment $Z(P)Z(H)\subseteq Z(P)\implies Z(H)\subseteq Z(P)$, contra trivial intersection.