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David Easdown

Publications and source records attributed to David Easdown.

4 recordsLinked to original sources
arXiv2017

Absorption of Direct Factors With Respect to the Minimal Faithful Permutation Degree of a Finite Group

The minimal faithful permutation degree $μ(G)$ of a finite group $G$ is the least nonnegative integer $n$ such that $G$ embeds in the symmetric group $\Sym(n)$. We prove that if $H$ is a group then $μ(G)=μ(G\times H)$ for some group $G$ then $H$ embeds in $A\times Q^k$ for some abelian group of odd order, some generalised quaternion $2$-group and some nonnegative integer $k$. As a consequence, $μ(G^{n+1})=μ(G^n)$ for some nonnegative integer $n$ if and only if $G$ is trivial.

math.GR↗
arXiv2014

The Smallest Faithful Permutation Degree for a Direct Product Obeying an Inequality Condition

The minimal faithful permutation degree $μ(G)$ of a finite group $G$ is the least nonnegative integer $n$ such that $G$ embeds in the symmetric group $\Sym(n)$. Clearly $μ(G \times H) \le μ(G) + μ(H)$ for all finite groups $G$ and $H$. Wright (1975) proves that equality occurs when $G$ and $H$ are nilpotent and exhibits an example of strict inequality where $G\times H$ embeds in $\Sym(15)$. Saunders (2010) produces an infinite family of examples of permutation groups $G$ and $H$ where $μ(G \times H) < μ(G) + μ(H)$, including the example of Wright's as a special case. The smallest groups in Saunders' class embed in $\Sym(10)$. In this paper we prove that 10 is minimal in the sense that $μ(G \times H) = μ(G) + μ(H)$ for all groups $G$ and $H$ such that $μ(G\times H)\le 9$.

math.GR↗
arXiv2007

A Presentation for the Dual Symmetric Inverse Monoid

The dual symmetric inverse monoid $\mathscr{I}_n^*$ is the inverse monoid of all isomorphisms between quotients of an $n$-set. We give a monoid presentation of $\mathscr{I}_n^*$ and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

math.GR↗