ON SOME COMBINATORIAL AND GROUP- THEORETIC STUDY OF WREATH PRODUCT ACTIONS

GROUP- THEORETIC STUDY OF WREATH PRODUCT ACTIONS

Abstract

Let $G \leq \mathrm{Sym}(X)$ be a finite permutation group acting faithfully and transitively on a set $X$ of size $m$, and let $W = G \wr S_n = G^n \rtimes S_n$ act in product action on the Cartesian power $X^n$. 

In this paper, we develop a systematic combinatorial and group-theoretic framework for analyzing the permutation structure of $W$ by reducing global invariants to the orbit structure of the point stabilizer $G_{x_0}$ in the base group. We prove that rank and subdegrees of $G \wr S_n$ are completely determined by multinomial distributions arising from the stabilizer orbit partition of $X$. Explicit closed formulas are obtained for rank, subdegrees, base size, minimal degree, and fixity. Furthermore, we establish connectivity of all non-diagonal orbital graphs in product action.

Our results demonstrate that the global permutation behaviour of wreath products in product action is governed entirely by stabilizer orbit geometry in the base group. Specialisations to symmetric, alternating, cyclic, and dihedral groups illustrate how local stabilizer structure dictates global combinatorial properties.