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Displaying similar documents to “Some results on Sylow numbers of finite groups”

On R -conjugate-permutability of Sylow subgroups

Xianhe Zhao, Ruifang Chen (2016)

Czechoslovak Mathematical Journal

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A subgroup H of a finite group G is said to be conjugate-permutable if H H g = H g H for all g G . More generaly, if we limit the element g to a subgroup R of G , then we say that the subgroup H is R -conjugate-permutable. By means of the R -conjugate-permutable subgroups, we investigate the relationship between the nilpotence of G and the R -conjugate-permutability of the Sylow subgroups of A and B under the condition that G = A B , where A and B are subgroups of G . Some results known in the literature are improved...

On σ -permutably embedded subgroups of finite groups

Chenchen Cao, Li Zhang, Wenbin Guo (2019)

Czechoslovak Mathematical Journal

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Let σ = { σ i : i I } be some partition of the set of all primes , G be a finite group and σ ( G ) = { σ i : σ i π ( G ) } . A set of subgroups of G is said to be a complete Hall σ -set of G if every non-identity member of is a Hall σ i -subgroup of G and contains exactly one Hall σ i -subgroup of G for every σ i σ ( G ) . G is said to be σ -full if G possesses a complete Hall σ -set. A subgroup H of G is σ -permutable in G if G possesses a complete Hall σ -set such that H A x = A x H for all A and all x G . A subgroup H of G is σ -permutably embedded in...

On solvability of finite groups with some s s -supplemented subgroups

Jiakuan Lu, Yanyan Qiu (2015)

Czechoslovak Mathematical Journal

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A subgroup H of a finite group G is said to be s s -supplemented in G if there exists a subgroup K of G such that G = H K and H K is s -permutable in K . 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 G is solvable if every subgroup of odd prime order of G is s s -supplemented in G , and that G is solvable if and only if every Sylow subgroup of odd order of G is s s -supplemented in G . These results...

Finite p -nilpotent groups with some subgroups weakly -supplemented

Liushuan Dong (2020)

Czechoslovak Mathematical Journal

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Suppose that G is a finite group and H is a subgroup of G . Subgroup H is said to be weakly -supplemented in G if there exists a subgroup B of G such that (1) G = H B , and (2) if H 1 / H G is a maximal subgroup of H / H G , then H 1 B = B H 1 < G , where H G is the largest normal subgroup of G contained in H . We fix in every noncyclic Sylow subgroup P of G a subgroup D satisfying 1 < | D | < | P | and study the p -nilpotency of G under the assumption that every subgroup H of P with | H | = | D | is weakly -supplemented in G . Some recent results are generalized. ...

On the conjugate type vector and the structure of a normal subgroup

Ruifang Chen, Lujun Guo (2022)

Czechoslovak Mathematical Journal

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Let N be a normal subgroup of a group G . The structure of N is given when the G -conjugacy class sizes of N is a set of a special kind. In fact, we give the structure of a normal subgroup N under the assumption that the set of G -conjugacy class sizes of N is ( p 1 n 1 a 1 n 1 , , p 1 1 a 11 , 1 ) × × ( p r n r a r n r , , p r 1 a r 1 , 1 ) , where r > 1 , n i > 1 and p i j are distinct primes for i { 1 , 2 , , r } , j { 1 , 2 , , n i } .

Finite groups whose all proper subgroups are 𝒞 -groups

Pengfei Guo, Jianjun Liu (2018)

Czechoslovak Mathematical Journal

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A group G is said to be a 𝒞 -group if for every divisor d of the order of G , there exists a subgroup H of G of order d such that H is normal or abnormal in G . We give a complete classification of those groups which are not 𝒞 -groups but all of whose proper subgroups are 𝒞 -groups.

The p -nilpotency of finite groups with some weakly pronormal subgroups

Jianjun Liu, Jian Chang, Guiyun Chen (2020)

Czechoslovak Mathematical Journal

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For a finite group G and a fixed Sylow p -subgroup P of G , Ballester-Bolinches and Guo proved in 2000 that G is p -nilpotent if every element of P G ' with order p lies in the center of N G ( P ) and when p = 2 , either every element of P G ' with order 4 lies in the center of N G ( P ) or P is quaternion-free and N G ( P ) is 2 -nilpotent. Asaad introduced weakly pronormal subgroup of G in 2014 and proved that G is p -nilpotent if every element of P with order p is weakly pronormal in G and when p = 2 , every element of P with...

A note on sumsets of subgroups in * p

Derrick Hart (2013)

Acta Arithmetica

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Let A be a multiplicative subgroup of * p . Define the k-fold sumset of A to be k A = x 1 + . . . + x k : x i A , 1 i k . We show that 6 A * p for | A | > p 11 / 23 + ϵ . In addition, we extend a result of Shkredov to show that | 2 A | | A | 8 / 5 - ϵ for | A | p 5 / 9 .

Every 2 -group with all subgroups normal-by-finite is locally finite

Enrico Jabara (2018)

Czechoslovak Mathematical Journal

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A group G has all of its subgroups normal-by-finite if H / H G is finite for all subgroups H of G . The Tarski-groups provide examples of p -groups ( p a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2 -group with every subgroup normal-by-finite is locally finite. We also prove that if | H / H G | 2 for every subgroup H of G , then G contains an Abelian subgroup of index at most 8 .

On a generalization of a theorem of Burnside

Jiangtao Shi (2015)

Czechoslovak Mathematical Journal

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A theorem of Burnside asserts that a finite group G is p -nilpotent if for some prime p a Sylow p -subgroup of G lies in the center of its normalizer. In this paper, let G be a finite group and p the smallest prime divisor of | G | , the order of G . Let P Syl p ( G ) . As a generalization of Burnside’s theorem, it is shown that if every non-cyclic p -subgroup of G is self-normalizing or normal in G then G is solvable. In particular, if P a , b | a p n - 1 = 1 , b 2 = 1 , b - 1 a b = a 1 + p n - 2 , where n 3 for p > 2 and n 4 for p = 2 , then G is p -nilpotent or p -closed. ...