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 .
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Derrick Hart (2013)
Acta Arithmetica
Similarity:
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...
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 , .
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. ...
Fenfang Xie, Jinjin Wang, Jiayi Xia, Guo Zhong (2013)
Confluentes Mathematici
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Let be a finite group, the smallest prime dividing the order of and a Sylow -subgroup of with the smallest generator number . There is a set of maximal subgroups of such that . In the present paper, we investigate the structure of a finite group under the assumption that every member of is either -permutably embedded or weakly -permutable in to give criteria for a group to be -supersolvable or -nilpotent.
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...
Paul Balmer (2013)
Journal of the European Mathematical Society
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Let be a field of characteristic . Let be a finite group of order divisible by and a -Sylow subgroup of . We describe the kernel of the restriction homomorphism , for the group of endotrivial representations. Our description involves functions that we call weak -homomorphisms. These are generalizations to possibly non-normal of the classical homomorphisms appearing in the normal case.