Characterizing Pyramidal Hadamard Designs With the Largest Number of Fixed Points

A symmetric $\left(v,k,\lambda \right)$-design is said to be $f$-pyramidal, with $f<v-1$, under the action of a group $G$ if $G$ acts as an automorphism group fixing $f$ points and acting sharply transitively on the remaining ones. We show that, necessarily, $f\le v-2\left(k-\lambda \right)$. In particular, for a Hadamard design with parameters $\left(v,k,\lambda \right)=\left(4m-1,2m-1,m-1\right)$, for some $m\in N$, it follows that $f\le 2m-1$. Recently, working on the complement design, the family $P$ of Hadamard $\left(4m-1,2m-1,m-1\right)$-designs admitting an abelian $\left(2m-1\right)$-pyramidal automorphism group $G$ has been completely determined in the case $m=2^k>1$. In this paper, we generalize that result by showing that $P$ coincides with the family of Hadamard $\left(4m-1,2m-1,m-1\right)$-designs admitting a $\left(2m-1\right)$-pyramidal automorphism group, without assuming either that the group is abelian or that $m$ is a power of 2.