Recognizing by Spectrum for the Automorphism Groups of Sporadic Simple Groups

Abstract

The spectrum of a finite group is the set of its element orders, and two groups are said to be isospectral if they have the same spectra. A finite group G is said to be recognizable by spectrum, if every finite group isospectral with G is isomorphic to G. We prove that if S is one of the sporadic simple groups $M^cL$, $M_{12}$, $M_{22}$, He, Suz and $O'N$, then $\mathrm{Aut}(S)$ is recognizable by spectrum. This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups, except $J_2$.

Keywords

Automorphism groups of sporadic simple groups / Prime graph / Recognizable by spectrum

References

[1] Gorenstein D. Finite Groups. 1980 2 New York: Chelsea Publishing Co.

[2] Kondratév AS. On prime graph components of finite simple groups. Mat. Sb.. 1989, 180 6 787-797

[3] Li H, Shi WJ. A characterization of some sporadic simple groups. Chin. Ann. Math.. 1993, 14A 2 144-151(in Chinese)

[4] Lucido MS, Moghaddamfar AR. Groups with complete prime graph connected components. J. Group Theory. 2004, 7 3 373-384

[5] Mazurov VD. Characterizations of finite groups by sets of element orders. Algebra Logic. 1997, 36 1 23-32

[6] Mazurov VD, Shi WJ. A note to the characterization of sporadic simple groups. Algebra Colloq.. 1998, 5 3 285-288

[7] Mazurov VD. Recognition of finite simple groups $S_4(q)$ by their element orders. Algebra Logic. 2002, 41 2 93-110

[8] Moghaddamfar AR, Zokayi AR, Darafsheh MR. On the characterizability of the automorphism groups of sporadic simple groups by their element orders. Acta Math. Sin.. 2004, 20 4 653-662

[9] Praeger CE, Shi WJ. A characterization of some alternating and symmetric groups. Commun. Algebra. 1994, 22 5 1507-1530

[10] Shi WJ. A charateristic property of $J_1$ and $\text{ PSL }_2(2^n)$. Adv. Math.. 1987, 16 397-401(in Chinese)

[11] Shi WJ. A charateristic property of Mathieu groups. Chin. Ann. Math.. 1988, 9A 5 575-580(in Chinese)

[12] Shi WJ. A charateristic property of Conway simple group $Co_2$. Chin. J. Math.. 1989, 9 2 171-172(in Chinese)

[13] Shi WJ. A characterization of the Higman-Sims simple group. Houston J. Math.. 1990, 16 597-602

[14] Shi WJ. The characterization of the sporadic simple groups by their element orders. Algebra Colloq.. 1994, 1 2 159-166

[15] Shi WJ, Li H. A charateristic property of $M_{12}$ and $\text{ PSU }(6,2)$. Acta Math. Sin.. 1989, 32 6 758-764(in Chinese)

[16] Vasilev AV. On recognition of all finite nonabelian simple groups with orders having prime divisors at most 13. Sib. Math. J.. 2005, 46 2 246-253

[17] Williams JS. Prime graph components of finite groups. J. Algebra. 1981, 69 2 487-513

[18] Zavarnitsine AV. Recognition of alternating group of degrees $r+1$ and $r+2$ for primes $r$ and the group of degree 16 by their element order sets. Algebra Logic. 2000, 39 6 370-377

Funding

Russian Science Foundation(14-21-00065)