| A generalized central trinomial coefficient Tn(b, c) is the coefficient of xn in the expansion of (x2+bx+c)n with b, c∈Z. Let p> 3 be a prime and let m∈Z with m≠0 (mod p). Recently, Z.W.Sun observed certain patterns concerning with (b,c,m)=(5,4,4), (3,1,4), (4,9,16), (8,25,16), (12,25,16). In Chapter 2, We will confirm Sun's observations.In Chapter 3, we deal with combinatorial aspects of finite covers of groups by subgroups. Let G1,..., Gk be subgroups of a group G such that{Gi}i=1k forms a minimal m-cover of G. In 2004, Z.W.Sun and his student S.Guo posed the follow-ing conjecture:If G1,..., Gk are subnormal subgroups of G with [G:∩i=1k Gi]=Πt=1r ptαt,where p1,...,pr are distinct primes andα1,...,αr are positive integers, then k> m+∑t=1r(αt-1)(pt-1).We will prove this conjecture for G=Cq⊕Cqn, where q is a prime or a product of two distinct primes, and n is a squarefree pos-itive integer coprime to q. |
