Characterization of finite groups by their commuting graph.
Iranmanesh, A., Jafarzadeh, A. (2007)
Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]
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Displaying similar documents to “Quasirecognition by prime graph of where is a prime.”
Characterization of finite groups by their commuting graph.
Uniqueness of Factoring an Integer and Multiplicative Group Z/pZ*
Hiroyuki Okazaki, Yasunari Shidama (2008)
Formalized Mathematics
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In the [20], it had been proven that the Integers modulo p, in this article we shall refer as Z/pZ, constitutes a field if and only if Z/pZ is a prime. Then the prime modulo Z/pZ is an additive cyclic group and Z/pZ* = Z/pZ{0is a multiplicative cyclic group, too. The former has been proven in the [23]. However, the latter had not been proven yet. In this article, first, we prove a theorem concerning the LCM to prove the existence of primitive elements of Z/pZ*. Moreover we prove the...
On recognition of the projective special linear groups over binary fields.
Grechkoseeva, M.A., Lucido, Maria Silvia, Mazurov, V.D., Moghaddamfar, Ali Reza, Vasil'ev, A.V. (2005)
Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]
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A new characterization of some alternating and symmetric groups.
Khosravi, Amir, Khosravi, Behrooz (2003)
International Journal of Mathematics and Mathematical Sciences
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Basic Properties of Primitive Root and Order Function
Na Ma, Xiquan Liang (2012)
Formalized Mathematics
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In this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on [12], [8], and [11].
On the prime graphs of the automorphism groups of sporadic simple groups
Behrooz Khosravi (2009)
Archivum Mathematicum
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In this paper as the main result, we determine finite groups with the same prime graph as the automorphism group of a sporadic simple group, except .
Properties of Primes and Multiplicative Group of a Field
Kenichi Arai, Hiroyuki Okazaki (2009)
Formalized Mathematics
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In the [16] has been proven that the multiplicative group Z/pZ* is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group.Meanwhile, in cryptographic system like RSA, in which security basis depends upon the difficulty...
Proth Numbers
Christoph Schwarzweller (2014)
Formalized Mathematics
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In this article we introduce Proth numbers and prove two theorems on such numbers being prime [3]. We also give revised versions of Pocklington’s theorem and of the Legendre symbol. Finally, we prove Pepin’s theorem and that the fifth Fermat number is not prime.
OD-characterization of alternating groupsA p + d
Yong Yang, Shitian Liu, Zhanghua Zhang (2017)
Open Mathematics
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Let An be an alternating group of degree n. Some authors have proved that A10, A147 and A189 cannot be OD-characterizable. On the other hand, others have shown that A16, A23+4, and A23+5 are OD-characterizable. We will prove that the alternating groups Ap+d except A10, are OD-characterizable, where p is a prime and d is a prime or equals to 4. This result generalizes other results.
The Perfect Number Theorem and Wilson's Theorem
Marco Riccardi (2009)
Formalized Mathematics
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This article formalizes proofs of some elementary theorems of number theory (see [1, 26]): Wilson's theorem (that n is prime iff n > 1 and (n - 1)! ≅ -1 (mod n)), that all primes (1 mod 4) equal the sum of two squares, and two basic theorems of Euclid and Euler about perfect numbers. The article also formally defines Euler's sum of divisors function Φ, proves that Φ is multiplicative and that Σk|n Φ(k) = n.