Let G be a finite group.We denote byν(G)the probability that two randomly chosen elements of G generate a nilpotent subgroup.In this paper,we characterize the structure of finite groups G with lower bounds\({1\over p...Let G be a finite group.We denote byν(G)the probability that two randomly chosen elements of G generate a nilpotent subgroup.In this paper,we characterize the structure of finite groups G with lower bounds\({1\over p},\,{{p^{2}+8}\over{9p^{2}}}\)and\({p+3}\over{4p}\)onν(G),where p is a prime divisor of∣G∣.展开更多
This paper considers such a group G which possesses nontrivial proper subgroups H 1 ,H 2 such that any proper subgroup of G not contained in H 1 ∪ H 2 is p-closed and obtains that if G is soluble,then the number of p...This paper considers such a group G which possesses nontrivial proper subgroups H 1 ,H 2 such that any proper subgroup of G not contained in H 1 ∪ H 2 is p-closed and obtains that if G is soluble,then the number of prime divisors contained in ｜G｜ is 2,3 or 4;if not,then it has a form x N where N/Φ（N） is a non-abelian simple group.Then the structure of such a group is determined for p = 2,H 1 = H 2 under some conditions.展开更多
基金Supported by NSF of China(Grant No.12061011)NSF of Guangxi(Grant Nos.2020GXNSFDA238014,2021GXNSFAA220105)。
文摘Let G be a finite group.We denote byν(G)the probability that two randomly chosen elements of G generate a nilpotent subgroup.In this paper,we characterize the structure of finite groups G with lower bounds\({1\over p},\,{{p^{2}+8}\over{9p^{2}}}\)and\({p+3}\over{4p}\)onν(G),where p is a prime divisor of∣G∣.
基金Supported by the Science Foundation of the Ministry of Education of China for the Returned Overseas Scholars (Grant No. 2008101)the Science Foundation of Shanxi Province for the Returned Overseas Scholars (Grant No. 200799)the Doctoral Science Foundation of Shanxi Datong University (Grant No. 2008-B-02)
文摘This paper considers such a group G which possesses nontrivial proper subgroups H 1 ,H 2 such that any proper subgroup of G not contained in H 1 ∪ H 2 is p-closed and obtains that if G is soluble,then the number of prime divisors contained in ｜G｜ is 2,3 or 4;if not,then it has a form x N where N/Φ（N） is a non-abelian simple group.Then the structure of such a group is determined for p = 2,H 1 = H 2 under some conditions.
