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Displaying similar documents to “On a generalization of a theorem of Burnside”

The -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 and a fixed Sylow -subgroup of , Ballester-Bolinches and Guo proved in 2000 that is -nilpotent if every element of with order lies in the center of and when , either every element of with order lies in the center of or is quaternion-free and is -nilpotent. Asaad introduced weakly pronormal subgroup of in 2014 and proved that is -nilpotent if every element of with order is weakly pronormal in and when , every element of with...

Finite -nilpotent groups with some subgroups weakly -supplemented

Liushuan Dong (2020)

Czechoslovak Mathematical Journal

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

A note on infinite -groups

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...

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 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 , .

On -permutably embedded subgroups of finite groups

Chenchen Cao, Li Zhang, Wenbin Guo (2019)

Czechoslovak Mathematical Journal

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Let be some partition of the set of all primes , be a finite group and . A set of subgroups of is said to be a complete Hall -set of if every non-identity member of is a Hall -subgroup of and contains exactly one Hall -subgroup of for every . is said to be -full if possesses a complete Hall -set. A subgroup of is -permutable in if possesses a complete Hall -set such that = for all and all . A subgroup of is -permutably embedded in...

On solvability of finite groups with some -supplemented subgroups

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...

On the derived length of units in group algebra

Dishari Chaudhuri, Anupam Saikia (2017)

Czechoslovak Mathematical Journal

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Let be a finite group , a field of characteristic and let be the group of units in . We show that if the derived length of does not exceed , then must be abelian.

Some results on Sylow numbers of finite groups

Yang Liu, Jinjie Zhang (2024)

Czechoslovak Mathematical Journal

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We study the group structure in terms of the number of Sylow -subgroups, which is denoted by . The first part is on the group structure of finite group such that , where is a normal subgroup of . The second part is on the average Sylow number and we prove that if is a finite nonsolvable group with and , then , where is the Fitting subgroup of . In the third part, we study the nonsolvable group with small sum of Sylow numbers.