Abstract:The order of every finite group $G$ can be expressed as a product of coprime positive integers $m_1,..., m_t$ such that $\pi (m_i)$ is a connected component of the prime graph of $G$. The integers $m_1,..., m_t$ are called the order components of $G$. Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups $C_2(q)$ where $q>5$ are also uniquely determined by their order components. As corollaries of this result, the validities of a conjecture by J.G. Thompson and a conjecture by W. Shi and J. Be for $C_2(q)$ with $q>5$ are obtained.
Keywords: prime graph, order component, finite group, simple group
AMS Subject Classification: 20D05, 20D60