Some Quantitative Characterizations of Certain Symplectic Groups

TL;DR

This paper proves that the symplectic group over a binary field with a Mersenne prime is uniquely determined by its degree pattern and order components, establishing its OD- and OC-characterizability.

Contribution

It demonstrates that certain symplectic groups over binary fields are uniquely characterized by their degree pattern and order components.

Findings

The symplectic group C_p(2) is OD-characterizable.

The symplectic group C_p(2) is OC-characterizable.

Uniqueness holds for groups with Mersenne prime conditions.

Abstract

Given a finite group $G$, denote by ${\rm D}(G)$ the degree pattern of $G$ and by ${\rm OC}(G)$ the set of all order components of $G$. Denote by $h_{{\rm OD}}(G)$ (resp. $h_{{\rm OC}}(G)$) the number of isomorphism classes of finite groups $H$ satisfying conditions $|H|=|G|$ and ${\rm D}(H)={\rm D}(G)$ (resp. ${\rm OC}(H)={\rm OC}(G)$). A finite group $G$ is called OD-characterizable (resp. OC-characterizable) if $h_{\rm OD}(G)=1$ (resp. $h_{\rm OC}(G)=1$). Let $C=C_p(2)$ be a symplectic group over binary field, for which $2^p-1>7$ is a Mersenne prime. The aim of this article is to prove that $h_{\rm OD}(C)=1=h_{\rm OC}(C)$.