A note on sumsets of subgroups in
Derrick Hart (2013)
Acta Arithmetica
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Let A be a multiplicative subgroup of . Define the k-fold sumset of A to be . We show that for . In addition, we extend a result of Shkredov to show that for .
Derrick Hart (2013)
Acta Arithmetica
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Let A be a multiplicative subgroup of . Define the k-fold sumset of A to be . We show that for . In addition, we extend a result of Shkredov to show that for .
Jiakuan Lu, Yanyan Qiu (2015)
Czechoslovak Mathematical Journal
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A subgroup of a finite group is said to be -supplemented in if there exists a subgroup of such that and is -permutable in . In this paper, we first give an example to show that the conjecture in A. A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group is solvable if every subgroup of odd prime order of is -supplemented in , and that is solvable if and only if every Sylow subgroup of odd order of is -supplemented in . These results...
Xianhe Zhao, Ruifang Chen (2016)
Czechoslovak Mathematical Journal
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A subgroup of a finite group is said to be conjugate-permutable if for all . More generaly, if we limit the element to a subgroup of , then we say that the subgroup is -conjugate-permutable. By means of the -conjugate-permutable subgroups, we investigate the relationship between the nilpotence of and the -conjugate-permutability of the Sylow subgroups of and under the condition that , where and are subgroups of . Some results known in the literature are improved...
Ruifang Chen, Lujun Guo (2022)
Czechoslovak Mathematical Journal
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Let be a normal subgroup of a group . The structure of is given when the -conjugacy class sizes of is a set of a special kind. In fact, we give the structure of a normal subgroup under the assumption that the set of -conjugacy class sizes of is , where , and are distinct primes for , .
Paolo Leonetti, Salvatore Tringali (2014)
Journal de Théorie des Nombres de Bordeaux
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Given an integer , let be pairwise coprime integers , a family of nonempty proper subsets of with “enough” elements, and a function . Does there exist at least one prime such that divides for some , but it does not divide ? We answer this question in the positive when the are prime powers and and are subjected to certain restrictions. We use the result to prove that, if and is a set of three or more primes that contains all prime divisors of any...
Reza Nikandish, Babak Miraftab (2015)
Czechoslovak Mathematical Journal
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Let be a group. If every nontrivial subgroup of has a proper supplement, then is called an -group. We study some properties of -groups. For instance, it is shown that a nilpotent group is an -group if and only if is a subdirect product of cyclic groups of prime orders. We prove that if is an -group which satisfies the descending chain condition on subgroups, then is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...
Azam Babai, Zeinab Akhlaghi (2017)
Czechoslovak Mathematical Journal
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Let be a group and be the set of element orders of . Let and be the number of elements of order in . Let nse. Assume is a prime number and let be a group such that nse nse, where is the symmetric group of degree . In this paper we prove that , if divides the order of and does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.
Václav Kryštof, Luděk Zajíček (2016)
Commentationes Mathematicae Universitatis Carolinae
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It is proved that real functions on which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower -functions, or of two strongly paraconvex functions) coincide with semismooth functions on (i.e. those locally Lipschitz functions on for which and for each ). Further, for each modulus , we characterize the class of functions on which can be written as , where and are semiconvex with modulus (for some ) using a new...
Jiangtao Shi (2015)
Czechoslovak Mathematical Journal
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A theorem of Burnside asserts that a finite group is -nilpotent if for some prime a Sylow -subgroup of lies in the center of its normalizer. In this paper, let be a finite group and the smallest prime divisor of , the order of . Let . As a generalization of Burnside’s theorem, it is shown that if every non-cyclic -subgroup of is self-normalizing or normal in then is solvable. In particular, if , where for and for , then is -nilpotent or -closed. ...
Lola Thompson (2014)
Journal de Théorie des Nombres de Bordeaux
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In this paper, we examine a natural question concerning the divisors of the polynomial : “How often does have a divisor of every degree between and ?” In a previous paper, we considered the situation when is factored in . In this paper, we replace with , where is an arbitrary-but-fixed prime. We also consider those where this condition holds for all .
Moshe Marcus, Laurent Véron (2004)
Journal of the European Mathematical Society
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Let be a bounded domain of class in N and let be a compact subset of . Assume that and denote by the maximal solution of in which vanishes on . We obtain sharp upper and lower estimates for in terms of the Bessel capacity and prove that is -moderate. In addition we describe the precise asymptotic behavior of at points , which depends on the “density” of at , measured in terms of the capacity .
Daniel J. Katz, Philippe Langevin (2015)
Acta Arithmetica
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We consider Weil sums of binomials of the form , where F is a finite field, ψ: F → ℂ is the canonical additive character, , and . If we fix F and d, and examine the values of as a runs through , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...
Reese Scott, Robert Styer (2013)
Journal de Théorie des Nombres de Bordeaux
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We consider , the number of solutions to the equation in nonnegative integers and integers , for given integers , , , and . When , we show that except for a finite number of cases all of which satisfy for each solution; when , we show that except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving solutions.