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A characterization of \(C_2(q)\) where \(q>5\). (English) Zbl 1068.20020
Summary: The order of every finite group \(G\) can be expressed as a product of coprime positive integers \(m_1,\dots,m_t\) such that \(\pi(m_i)\) is a connected component of the prime graph of \(G\). The integers \(m_1,\dots,m_t\) 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. Bi for \(C_2(q)\) with \(q>5\) are obtained.

20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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