Abstract
In Li and Chen (Sib. Math. J. 53(2), 243–247, 2012), it is proved that the simple group \(A_1(p^n)\) is uniquely determined by the set of orders of its maximal abelian subgroups. Let \(q=p^{\alpha }\) be a prime power and \(L=A_2(q)\). In this paper, we prove that if \(q\) is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as \(L\), is isomorphic to \(L\) or an extension of \(L\) by a subgroup of the outer automorphism group of \(L\).
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Acknowledgments
The authors express their gratitude to Professor G. Malle for serious comments in the proof of Theorem 3.4 and Professor Nick Gill for very useful guidance in the proof of Theorem 3.9. Also the authors would like to thank the referee for invaluable comments and suggestions. The second author would like to thank Institute for Research in Fundamental Sciences (IPM).
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Communicated by J. S. Wilson.
B. Khosravi was supported in part by a grant from IPM (Institute for Research in Fundamental Sciences) (No. 91050116).
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Momen, Z., Khosravi, B. Groups with the same orders of maximal abelian subgroups as \(A_2(q)\) . Monatsh Math 174, 285–303 (2014). https://doi.org/10.1007/s00605-013-0510-5
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DOI: https://doi.org/10.1007/s00605-013-0510-5