A question on the Theory of Groups of Finite Order by Burnside There is a typical exercise in the book of Burnside above which I have totally no idea. Any hint would be appreciate.
Prove that if G is a non abelian group of order $p^m$ in which no conjugate set ( In common sense I think means conjugacy class ) contains more than p elements and if H is the centre of G, then G/H must have an even number of generators. Show also that the order of the derived group of G is p.
 A: If conjugate set indeed means conjugacy class, then first notice that if every class has size at most $p$, then for any $x\in G$, $C_G(x)$ has size at least $p^{m-1}$, hence either $G$ or maximal. The center $H$ is the intersection of all centralizers, thus the Frattini subgroup $\phi(G)$ is contained in $H$. In particular, $G/H$ is elementary abelian. The question is whether $n=\dim(G/H)$ (as a vector space over $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$) is even or odd.
For groups of class $2$, the map $$
\begin{split}
B:G/H \times G/H &\rightarrow [G,G]\\
(gH,hH)\mapsto [g,h]
\end{split}
$$ is bilinear, i.e., $(xyH,zH)\mapsto [x,z][y,z]$ and similarly $(xH,yzH)\mapsto [x,y][x,z]$.
There are two properties of $B$: Firstly, $B(v,v)=0$ for all $v\in G/H$ and secondly, $v^\perp=\{w\in G/H\mid B(v,w)=0\}$ has dimension $n-1$. The second property follows by our initial assumption that for $g\in G\setminus Z(G)$, we have $|C_G(g)|=p^{m-1}$. Can you show that these two properties imply that $\dim(G/H)$ is even? You need to use the fact that $B$ pairs each $v$ with $(G/H)/v^{\perp}$ and no vector can be paired with itself.
