A new characterization of $A_{p}$ with $p $ and $p-2$ are twin primes
Abstract
Let $G$ be a finite group and $\pi_{e}(G)$ be the set of elements order of $G$. Let $k \in \pi_{e}(G)$ and $m_{k}$ be the number of elements of order $k$ in $G$. Set nse($G$):=$\{ m_{k} | k \in \pi_{e}(G)\}$. Assume $p$ and $p-2$ are twin primes. We prove that if $G$ is a group such that nse($G$)=nse($A_{p}$) and $p\in \pi (G)$, then $G \cong A_{p}$. As a consequence of our results we prove that $A_{p}$ is uniquely determined by its nse and order.
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References
A. Iranmanesh, S. H. Alavi and B. Khosravi, A Characterization of PSL(3, q) where q is an odd prime power, J. Pure Appl. Algebra, 170, 243-254 (2002).
A. K. Asboei, S. S. salehi Amiri, A. Iranmanesh and A. Tehranian, A characterization of Symmetric group Sr, where r is prime number, Ann. Math. Inform, 40, 13-23 (2012).
A. K. Asboei, S. S. salehi Amiri, A. Iranmanesh and A. Tehranian, A new characterization of sporadic simple groups by NSE and order, J. Algebra Appl, 12(2), DOI: 10.1142/S0219498812501587 (2013).
A. K. Asboei, S. S. salehi Amiri, A. Iranmanesh and A. Tehranian, A new characterization of A7 and A8, An. Stint. Univ. Ovidius Constanta, 21(3), 43-50 (2013).
A. S. Kondtratev and V. D. Mazurove, Recognition Of Alternating groups of prime degree from their element orders, Siberian Mathematical Journal, 41(2), 294-302 (2000).
C. G. Shao, W. Shi and Q. H. Jiang, Characterization of simple K4-groups, Front Math China, 3, 355-370 (2008)
C. G. Shao and Q. H. Jiang, A new characterization of some linear groups by NSE, J. Alg. Appl, 13(2), DOI:10.1142/S0219498813500941 (2014).
D. J. S. Robinson, A course on the theory of groups, Springer-Verlag, New York, (1982).
G. Frobenius, Verallgemeinerung des sylowschen satze, Berliner sitz, 981-993 (1895).
G. Miller, Addition to a theorem due to Frobenius, Bull. Am. Math. Soc, 11, 6-7 (1904).
G. Y. Chen, On structure of Frobenius and 2-Frobenius group, Journal of Southwest China Normal University, 20(5), 485-487, (1995)(Chinese).
G. Y. Chen, On Thompson's conjecture, J. Algebra, 185(1), 184-193 (1996).
G. Y. Chen, Further reflections on Thompson’s conjecture, J. Algebra, 218, 276-285 (1999).
H. Zassenhaus, The theory of groups, 2nd ed. Chelsea Publishing Company, New York, (1958).
J. Bi, Characteristic of Alternating Groups by Orders of Normalizers of Sylow Subgroups, Algebra Colloq, 8(3), 249-256 (2001).
J. H. Conway, R. T. Curtis, S. P. Norton and R. A. Wilson, Atlas of Finite Groups, Clarendon, Oxford, (1985).
J. S. Williams, Prime graph components of finite groups, J. Algebra, 69(2), 487-513 (1981).
N. Iiyori and H. Yamaki, Prime graph components of the simple groups of Lie type over the field of even characteristic, J. Algebra, 155(2), 335-343 (1993).
R. Shen, C. G. Shao, Q. Jiang, W. Shi and V. Mazuro, A New Characterization of A5, Monatsh Math, 337-341 (2010).
V. D. Mazurov and E. I. Khukhro, Unsolved Problems in group theory, the Kourovka Notebook, 16 ed, Novosibirsk, Inst. Mat. Sibirsk. Otdel. Akad, (2006).
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