A new characterization of the automorphism groups of Mathieu groups

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">cd</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\rm{cd}}\left(G) be the set of irreducible complex character degrees of a finite group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G . <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ρ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \rho \left(G) denotes the set of primes dividing degrees in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">cd</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\rm{cd}}\left(G) . For any prime p , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mi>max</m:mi> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>χ</m:mi> <m:msub> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mspace width="0.08em" /> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow /> </m:mrow> <m:mspace width="0.08em" /> <m:mi>χ</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">Irr</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> {p}^{{e}_{p}\left(G)}=\max \left\{\chi {\left(1)}_{p}\hspace{0.08em}| \hspace{0.08em}\chi \in {\rm{Irr}}\left(G)\right\} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mspace width="0.08em" /> <m:mrow> <m:mo stretchy="false">∣</m:mo> <m:mrow /> </m:mrow> <m:mspace width="0.1em" /> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mi>ρ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> V\left(G)=\left\{{p}^{{e}_{p}\left(G)}\hspace{0.08em}| \hspace{0.1em}p\in \rho \left(G)\right\} . The degree prime-power graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \Gamma \left(G) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G is a graph whose vertices set is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> V\left(G) , and two vertices <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>∈</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> x,y\in V\left(G) are joined by an edge if and only if there exists <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="normal">cd</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> m\in {\rm{cd}}\left(G) such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mi>y</m:mi> <m:mo>∣</m:mo> <m:mi>m</m:mi> </m:math> xy| m . It is an interesting and difficult problem to determine the structure of a finite group by using its degree prime-power graphs. Qin proved that all Mathieu groups can be uniquely determined by their orders and degree prime-power graphs. In this article, we continue this topic and successfully characterize all the automorphism groups of Mathieu groups by using their orders and degree prime-power graphs.