Orbit Structures of Automorphism Groups of Designs

Abstract

In the last two decades, mathematicians have discussed various transivities of automorphism groups of designs (i.e., point, block, and flag transivities), from all these studies, we know that
$$0\le O^{\mathrm{\#}}(G,\mathbf{B})–O^{\mathrm{\#}}(G,\mathbf{X})\le |\mathbf{B}|–|\mathbf{X}|$$
for $2-(v,k,\lambda )$ designs (see $[$BMP\]). In this paper, we discuss the orbit structure of general combinatorial designs $\mathbf{D}(\mathbf{X},\mathbf{B})$ and obtain the equalities $$O^{\mathrm{\#}}(G,\mathbf{F})=\sum _{i=1}^u{O}^{\mathrm{\#}}(H(x_i),X_i)=\sum _{j=1}^l{O}^{\mathrm{\#}}(H(B_j),B_j),$$
where $H(x_i)$ and $H(B_j)$ are the stabilizers of the point $x_i$ and the block $B_j$ respectively, $u=O^{\mathrm{\#}}(G,\mathbf{X})$, $l=O^{\mathrm{\#}}(G,\mathbf{B})$.