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A Characterization of PGL$(2,p^n)$ by Some Irreducible Complex Character Degrees
Publications de l'Institut Mathématique, _N_S_99 (2016) no. 113, p. 257

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For a finite group $G$, let $\operatorname{cd}(G)$ be the set of irreducible complex character degrees of $G$ forgetting multiplicities and $X_1(G)$ be the set of all irreducible complex character degrees of $G$ counting multiplicities. Suppose that $p$ is a prime number. We prove that if $G$ is a finite group such that $|G|=|\operatorname{PGL}(2,p)|$, $p\in\operatorname{cd}(G)$ and $\max(\operatorname{cd}(G))=p+1$, then $G\cong\operatorname{PGL}(2,p),~SL(2,p)$ or $\operatorname{PSL}(2,p)\times A$, where $A$ is a cyclic group of order $(2,p-1)$. Also, we show that if $G$ is a finite group with $X_1(G)=X_1(\operatorname{PGL}(2,p^n))$, then $G\cong\operatorname{PGL}(2,p^n)$. In particular, this implies that $\operatorname{PGL}(2,p^n)$ is uniquely determined by the structure of its complex group algebra.
DOI : 10.2298/PIM150111017H
Classification : 20C15 20E99
Keywords: irreducible character degree, classification theorem of the finite simple group, complex group algebras 
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     author = {Somayeh Heydari and Neda Ahanjideh},
     title = {A {Characterization} of {PGL}$(2,p^n)$ by {Some} {Irreducible} {Complex} {Character} {Degrees}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {257 },
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Somayeh Heydari; Neda Ahanjideh. A Characterization of PGL$(2,p^n)$ by Some Irreducible Complex Character Degrees. Publications de l'Institut Mathématique, _N_S_99 (2016) no. 113, p. 257 . doi: 10.2298/PIM150111017H

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