Some chains of inequalities of Hua-Wang type are established. These interesting results are the generalizations of some known inequalities. One of the new methods establishing inequalities is based on Sandor's goo...Some chains of inequalities of Hua-Wang type are established. These interesting results are the generalizations of some known inequalities. One of the new methods establishing inequalities is based on Sandor's good idea; another is applying the characteristics of nonlinear positive functionals.展开更多
Let G be a group of order pn, p a prime. For 0 m n, sm(G) denotes the number of subgroups of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan had ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(...Let G be a group of order pn, p a prime. For 0 m n, sm(G) denotes the number of subgroups of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan had ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1+p, 1+p+p2 or 1+p+2p2(mod p3). The conjecture has a negative answer. In this paper, we further investigate the conjecture and propose two new conjectures.展开更多
文摘Some chains of inequalities of Hua-Wang type are established. These interesting results are the generalizations of some known inequalities. One of the new methods establishing inequalities is based on Sandor's good idea; another is applying the characteristics of nonlinear positive functionals.
基金supported by National Natural Science Foundation of China (Grant No.11071150)Natural Science Foundation of Shanxi Province (Grant No. 2008012001)The Returned Abroad-student Foundation of Shanxi Province (Grant No. [2007]13-56)
文摘Let G be a group of order pn, p a prime. For 0 m n, sm(G) denotes the number of subgroups of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan had ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1+p, 1+p+p2 or 1+p+2p2(mod p3). The conjecture has a negative answer. In this paper, we further investigate the conjecture and propose two new conjectures.
