On Alternating and Symmetric Groups Which Are Quasi OD-Characterizable

TL;DR

This paper investigates the OD-characterizability of symmetric and alternating groups, revealing that some are uniquely determined by their degree pattern while others have multiple non-isomorphic counterparts.

Contribution

It demonstrates that the symmetric group S_{27} is 38-fold OD-characterizable and identifies infinite families of such groups with high OD-characterizability.

Findings

S_{27} is 38-fold OD-characterizable

Existence of infinite families with k > 3

Multiple non-isomorphic groups share the same degree pattern

Abstract

Let $\Gamma(G)$ be the prime graph associated with a finite group $G$ and $D(G)$ be the degree pattern of $G$. A finite group $G$ is said to be $k$-fold OD-characterizable if there exist exactly $k$ non-isomorphic groups $H$ such that $|H|=|G|$ and $D(H)=D(G)$. The purpose of this article is twofold. First, it shows that the symmetric group $S_{27}$ is $38$-fold OD-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, $\{A_n\}$ and $\{S_n\}$, which are $k$-fold OD-characterizable with $k>3$.