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Let $G$ be a finite group. Let $X_1\left(G\right)$ be the first column of the ordinary character table of $G$. We will show that if $X_1\left(G\right)=X_1\left({\mathrm{PGU}}_3\left(q^2\right)\right)$, then $G\cong {\mathrm{PGU}}_3\left(q^2\right)$. As a consequence, we show that the projective general unitary groups ${\mathrm{PGU}}_3\left(q^2\right)$ are uniquely determined by the structure of their complex group algebras.

The order of every finite group $G$ can be expressed as a product of coprime positive integers $m_1,\cdots ,m_t$ such that $\pi \left(m_i\right)$ is a connected component of the prime graph of $G$. The integers $m_1,\cdots ,m_t$ 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 $C_2\left(q\right)$ where $q>5$ are also uniquely determined by their order components. As corollaries of this result, the validities of a...

In this paper, we consider finite groups with precisely one nonlinear nonfaithful irreducible character. We show that the groups of order 16 with nilpotency class 3 are the only $p$-groups with this property. Moreover we completely characterize the nilpotent groups with this property. Also we show that if $G$ is a group with a nontrivial center which possesses precisely one nonlinear nonfaithful irreducible character then $G$ is solvable.

Let $G$ be a finite group and ${\pi }_e\left(G\right)$ be the set of element orders of $G$. Let $k\in {\pi }_e\left(G\right)$ and $m_k$ be the number of elements of order $k$ in $G$. Set $\mathrm{nse}\left(G\right):=\{m_k:k\in {\pi }_e\left(G\right)\}$. In fact $\mathrm{nse}\left(G\right)$ is the set of sizes of elements with the same order in $G$. In this paper, by $\mathrm{nse}\left(G\right)$ and order, we give a new characterization of finite projective special linear groups $L_2\left(p\right)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $\left|G\right|=|L_2\left(p\right)|$ and $\mathrm{nse}\left(G\right)$ consists of $1$, $p^2-1$, $p(p+ϵ)/2$ and some numbers divisible by $2p$, where $p$ is a prime greater than...

Let $G$ be a finite group and let $N\left(G\right)$ denote the set of conjugacy class sizes of $G$. Thompson’s conjecture states that if $G$ is a centerless group and $S$ is a non-abelian simple group satisfying $N\left(G\right)=N\left(S\right)$, then $G\cong S$. In this paper, we investigate a variation of this conjecture for some symmetric groups under a weaker assumption. In particular, it is shown that $G\cong \mathrm{Sym}(p+1)$ if and only if $\left|G\right|=(p+1)!$ and $G$ has a special conjugacy class of size $(p+1)!/p$, where $p>5$ is a prime number. Consequently, if $G$ is a centerless group with $N\left(G\right)=N\left(\mathrm{Sym}\right(p+1\left)\right)$, then $G\cong \mathrm{Sym}(p+1)$.

In this note connected, edge-transitive lexicographic and Cartesian products are characterized. For the lexicographic product G ◦ H of a connected graph G that is not complete by a graph H, we show that it is edge-transitive if and only if G is edge-transitive and H is edgeless. If the first factor of G ∘ H is non-trivial and complete, then G ∘ H is edge-transitive if and only if H is the lexicographic product of a complete graph by an edgeless graph. This fixes an error of Li, Wang, Xu, and Zhao...