The Action of the Permutation Group (G, X) on the Power Set Ƥ(X)
Year: 2026
Type: article
Abstract: In this paper the main purpose is to investigate some properties of the permutation group \((G, X)\) acting on the \(\mathrm{P}(\mathrm{X})\). Harary (1969) investigated the action of the symmetric group \(S_n\) on unordered pairs \(X^{(2)}\) of points from the set \(X=\{1,2, \ldots, n\}\). he was able to calculate the number of graphs on \(n\) vertices. Palmer (1973) extended the work of Harary by investigating the action of the symmetric group \(S_n\) on unordered r-element subsets \(\mathrm{X}^{(\mathrm{r})}\) and calculated the number of r plexes using the formula \(|F i x \overline{(g)}|=\sum_i \prod_k\binom{\alpha_k}{i_k}\). Nyaga (2012) showed that if \(G=S_n\) acts on \(A^{(r)}\) then, \(\left|\operatorname{Stab}_G\{1,2, \ldots, r\}\right|=(n-r)!r!\).Jinbao et al. (2025) proved that there is no permutation group of degree \(\underline{\underline{\mathrm{n}}}>\mathrm{r} \geq 11\) that will have \(\mathrm{n}+\mathrm{r}\) setorbit.This work extends the Palmer's work by investigating the action of \(S_n\) on the \(\mathrm{P}(X)\); the set of all subsets of \(X\) called the power set of X , where \(X=\{1,2, \ldots, n\}\).
Author: M. P. Mungai
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7148702089
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