Displaying similar documents to “On solvability of finite groups with some s s -supplemented subgroups”

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 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 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. ...

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 p -subgroups, which is denoted by n p ( G ) . The first part is on the group structure of finite group G such that n p ( G ) = n p ( G / N ) , where N is a normal subgroup of G . The second part is on the average Sylow number asn ( G ) and we prove that if G is a finite nonsolvable group with asn ( G ) < 39 / 4 and asn ( G ) 29 / 4 , then G / F ( G ) A 5 , where F ( G ) is the Fitting subgroup of G . In the third part, we study the nonsolvable group with small sum of Sylow numbers.

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. ...

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...

On the derived length of units in group algebra

Dishari Chaudhuri, Anupam Saikia (2017)

Czechoslovak Mathematical Journal

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Let G be a finite group G , K a field of characteristic p 17 and let U be the group of units in K G . We show that if the derived length of U does not exceed 4 , then G must be abelian.

A note on infinite a S -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

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Let G be a group. If every nontrivial subgroup of G has a proper supplement, then G is called an a S -group. We study some properties of a S -groups. For instance, it is shown that a nilpotent group G is an a S -group if and only if G is a subdirect product of cyclic groups of prime orders. We prove that if G is an a S -group which satisfies the descending chain condition on subgroups, then G is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...